#72300929

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 59796 conjugacy classes of subgroups; only the 29668 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^11|c^2=hj,b^a=bh,c^a=chj,c^b=chj,d^a=di,d^b=dhijk,d^c=dk,e^a=ehj,e^c=ehj,f^a=fi,f^b=fhijk,f^c=fjk,f^e=fhijk,g^b=ghj,g^c=ghj,g^d=gi,g^e=ghj

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);