#72285839

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2446 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=ghi,e^2=ghij,b^a=bi,c^a=ch,c^b=cg,d^b=df,d^c=dfhij,e^a=ef,e^b=ef,e^c=eghij,e^d=ehi,f^c=fi,f^e=fij,g^a=gij,g^c=gij,g^d=gj,g^e=gj,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044
maximal normal subgroup of	#72286044

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