#72337233

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

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

4616782 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^11|a^2=k,g^2=k,b^a=bk,d^c=dk,g^a=gh,g^b=gi,g^c=gj,g^d=gi,g^e=gijk,g^f=ghk

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72337247, #72337394

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