#72334943

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

4627828 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:

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

Polycyclic presentation:

2^11|c^b=cj,d^a=dj,d^c=dj,g^b=gk,g^c=gh,g^d=gi,g^e=gh,g^f=gj

Conjugacy classes & character table

368 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72334985, #72335080

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