#72263254

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

17716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=h,f^2=ij,b^a=bh,c^a=ch,d^a=dj,d^b=dg,e^b=eh,e^c=ei,f^a=fh,f^c=fgj,f^d=fj,f^e=fij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263255, #72263317
maximal normal subgroup of	#72263255, #72263317

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