#72336751

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

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 65425 conjugacy classes of subgroups; only the 32128 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^11|f^2=hjk,g^2=hij,b^a=bh,c^b=chi,f^a=fhk,f^c=fj,f^d=fij,f^e=fhik,g^a=gk,g^c=ghij,g^d=ghj,g^e=gi

Conjugacy classes & character table

224 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, 6, F.8*F.10*F.11);;
gap> SetPower(coll, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);