#72336517

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 9324 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:

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

Polycyclic presentation:

2^10|a^2=g,d^2=h,f^2=ghi,b^a=bh,c^a=cgj,c^b=ci,d^a=dgh,d^c=dgj,e^a=eh,e^d=eh,f^a=fj,f^c=fgj,f^d=fhij,f^e=fgj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336654
maximal normal subgroup of	#72336654

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