#72336816

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

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 9712 conjugacy classes of subgroups; only the 3069 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=hj,f^2=ghi,b^a=bg,c^a=cghi,c^b=cghj,d^a=dh,d^b=dhj,d^c=dghi,f^a=fj,f^c=fghi,f^d=fgj,f^e=fhij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336887
maximal normal subgroup of	#72336887

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