#72333332

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 20089 conjugacy classes of subgroups; only the 5312 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^11|c^2=j,d^2=i,b^a=bk,c^b=ci,d^b=dg,d^c=dij,e^c=egi,e^d=egi,f^a=fj,f^b=fh,f^c=fj,f^d=fh

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333354
maximal normal subgroup of	#72333354

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