#72328051

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 22971 conjugacy classes of subgroups; only the 32280 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:

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

Polycyclic presentation:

2^11|b^a=bi,c^b=ch,f^b=fj,f^c=fk,f^d=fj,f^e=fik,g^a=gk,g^b=gj,g^c=gk,g^d=gj,g^e=ghik

Conjugacy classes & character table

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