#72329448

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 68759 conjugacy classes of subgroups; only the 32272 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)(13,14)(15,18)(16,17)(19,20);(14,17)(15,19)(22,25)(23,27);(21,22)(23,26)(24,25)(27,28);(6,9)(10,12)(15,19)(18,20)(23,27)(26,28);(2,4)(23,27)(26,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^11|f^2=j,g^2=ik,b^a=bk,c^b=ci,f^a=fj,f^b=fik,f^d=fj,f^e=fh,g^a=gj,g^b=gik,g^d=gik,g^e=gi

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72331654

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