#72332480

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

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 22494 conjugacy classes of subgroups; only the 39104 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^11|c^b=ch,f^b=fh,f^c=fk,f^d=fj,f^e=fh,g^a=gk,g^c=gk,g^d=gi

Conjugacy classes & character table

284 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72333354, #72336510

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