#72341084

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,7,13)(3,10,21,45)(4,25,29,50)(5,32,34,59)(6,36,38,49)(8,16,27,14)(9,17,18,58)(11,48,44,31)(12,28,53,19)(15,33,43,42)(20,63,37,51)(22,52,24,46)(23,47,35,57)(26,60,40,55)(30,61,62,56)(39,64,41,54);(1,3,18,34)(2,10,43,57)(4,26,62,40)(5,27,21,42)(6,30,38,29)(7,22,9,23)(8,24,33,35)(11,49,44,55)(12,25,64,50)(13,52,15,32)(14,45,17,47)(16,46,58,59)(19,31,28,20)(36,51,60,63)(37,39,48,41)(53,56,54,61);(1,4,27,48)(2,11,13,56)(3,19,21,38)(5,6,34,28)(7,20,8,30)(9,31,33,29)(10,12,52,60)(14,50,16,51)(15,61,43,44)(17,25,58,63)(18,62,42,37)(22,26,24,41)(23,40,35,39)(32,36,57,64)(45,49,46,54)(47,55,59,53);(1,5,33,35)(2,10,13,57)(3,7,22,42)(4,28,48,19)(6,37,38,62)(8,23,18,34)(9,24,27,21)(11,49,63,12)(14,59,16,46)(15,32,43,52)(17,47,58,45)(20,41,30,39)(25,64,56,55)(26,29,40,31)(36,61,53,50)(44,54,51,60);(1,6,7,28)(2,12,43,53)(3,20,35,31)(4,23,62,21)(5,29,22,30)(8,40,27,39)(9,41,18,26)(10,44,52,25)(11,47,50,59)(13,49,15,36)(14,60,17,55)(16,54,58,64)(19,33,38,42)(24,48,34,37)(32,56,57,51)(45,61,46,63)

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=gi,d^2=fhij,e^2=fh,b^a=bfg,c^a=ch,c^b=cg,d^a=dfghj,d^b=dfghi,d^c=dfij,e^a=ehij,e^b=efj,e^c=eghj,e^d=efhj,f^b=fi,f^c=fij,f^d=fj,f^e=fij,g^a=gi,g^d=gij,g^e=gi,h^a=hij,h^b=hj,h^c=hi,h^d=hj,h^e=hij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341091, #72341113
maximal normal subgroup of	#72341091, #72341113

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.7*F.10);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);