#72232630

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 9733 conjugacy classes of subgroups; only the 3077 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)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18)(19,21)(20,23)(22,24)(25,26,29,27)(28,31,32,30);(9,11)(10,14)(12,15)(13,16)(28,32)(30,31);(2,5)(3,8)(10,12)(11,16)(17,18,21,19)(20,22,24,23)(25,26)(27,29)(28,30)(31,32);(3,8)(6,7)(18,19)(20,24)(26,27)(28,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,18)(20,23,24,22)(25,27,29,26)(28,31,32,30);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^10|a^2=h,c^2=gh,e^2=j,f^2=gj,c^a=cg,c^b=ci,d^a=dj,d^c=dj,e^a=ehj,e^c=eh,e^d=ej,f^a=fgi,f^b=fghj,f^c=fgj,f^d=fgi,f^e=fgj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710
maximal normal subgroup of	#72232710

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