#72307182

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25599
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403908
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=efghij,g^2=ij,h^2=j,i^2=j,b^a=bf,d^a=dg,d^b=dghj,d^c=de,e^c=ej,e^d=ej,f^d=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307305
maximal normal subgroup of	#72307305

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