#72329124

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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 9968 conjugacy classes of subgroups; only the 3033 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)(3,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,32)(29,31);(10,15)(11,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,28)(26,30,31,32);(10,15)(12,14)(18,22)(20,23)(26,31)(27,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,30)(29,31);(1,2)(3,5)(4,7)(6,8)(9,12,13,14)(10,11,15,16)(17,20,21,23)(18,24,22,19)(25,28,29,27)(26,32,31,30)

Polycyclic presentation:

2^10|c^2=h,f^2=hi,b^a=bj,c^a=ci,c^b=cg,d^a=di,d^b=dh,d^c=dh,e^c=egh,e^d=egh,f^d=fhi,f^e=fhj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329242
maximal normal subgroup of	#72329242
maximal quotient of	#72331213

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