#72341045

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

2452 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)(3,12)(4,23)(5,31)(6,14)(7,15)(8,16)(9,21)(10,26)(11,33)(13,38)(17,58)(18,43)(19,54)(20,42)(22,55)(24,53)(25,47)(27,51)(28,50)(29,49)(30,48)(32,40)(34,52)(35,46)(36,56)(37,44)(39,57)(41,62)(45,61)(59,64)(60,63);(1,3,6,19)(2,9,14,43)(4,24,27,36)(5,30,33,26)(7,21,25,18)(8,22,38,20)(10,46,48,52)(11,50,31,63)(12,47,54,15)(13,44,16,45)(17,59,29,32)(23,40,51,64)(28,35,60,34)(37,39,61,62)(41,42,57,55)(49,53,58,56);(1,4,25,17)(2,10,47,63)(3,5,18,34)(6,27,7,29)(8,30,62,28)(9,40,54,56)(11,44,46,55)(12,53,43,64)(13,23,57,58)(14,48,15,50)(16,51,41,49)(19,33,21,35)(20,24,61,59)(22,36,37,32)(26,39,60,38)(31,45,52,42);(1,5,6,24)(2,11,13,31)(3,17,37,4)(7,35,25,32)(8,36,38,33)(9,23,54,50)(10,12,49,43)(14,56,16,53)(15,52,41,46)(18,30,22,28)(19,60,61,26)(20,29,21,27)(34,39,59,62)(40,47,64,57)(42,58,45,48)(44,51,55,63);(1,3,6,37)(2,12,13,43)(4,24,17,5)(7,21,25,20)(8,22,38,18)(9,41,54,15)(10,31,49,11)(14,44,16,55)(19,39,61,62)(23,52,50,46)(26,59,60,34)(27,35,29,32)(28,36,30,33)(40,58,64,48)(42,57,45,47)(51,56,63,53)

Polycyclic presentation:

2^10|b^2=g,c^2=gi,d^2=fgh,e^2=fgh,b^a=bi,c^a=cgj,c^b=cgj,d^a=dg,d^b=dh,d^c=df,e^a=eghj,e^b=efh,e^c=ehij,e^d=efgh,f^a=fj,f^b=fi,f^c=fj,f^d=fi,f^e=fj,g^d=gj,g^e=gij,h^b=hj,h^c=hij,h^d=hij,h^e=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341133, #72341188
maximal normal subgroup of	#72341133, #72341188

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, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.7*F.8);;
gap> SetPower(coll, 5, F.6*F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);