#72303667

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

13042 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:

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

Polycyclic presentation:

2^10|a^2=g,d^2=j,c^b=cgh,d^a=dh,e^a=eh,e^b=ehi,f^a=fh,f^c=fij,f^e=fj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303693
maximal normal subgroup of	#72303693

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