#72307169

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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=e,d^2=fj,e^2=i,g^2=ij,h^2=j,i^2=j,b^a=bfj,c^a=ch,c^b=cg,d^c=deg,e^a=ej,e^b=ei,e^d=ej,f^c=fj,g^a=gj,g^b=gij,g^c=gj,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.6);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.9);;
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*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);