#72336419

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 10325 conjugacy classes of subgroups; only the 3041 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,7)(3,4,8,6)(10,13)(11,15)(17,18,20,22)(19,23,24,21)(26,29)(30,31);(1,3)(2,6)(4,7)(5,8)(25,26)(27,31)(28,29)(30,32);(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(2,7)(4,6)(9,10,12,13)(11,16,15,14)(17,18,20,22)(19,23,24,21)(25,27)(26,31)(28,32)(29,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,16)(12,13)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,32)(29,30);(1,4)(2,8)(3,7)(5,6)(9,11)(10,14)(12,15)(13,16)(17,19,20,24)(18,21,22,23)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^10|a^2=h,d^2=ij,f^2=ghi,b^a=bg,c^a=cghi,c^b=ci,d^a=dhij,d^c=dghi,e^a=eij,e^d=eij,f^a=fj,f^c=fghi,f^d=fgij,f^e=fhj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336494
maximal normal subgroup of	#72336494

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