#72234898

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 46761 conjugacy classes of subgroups; only the 418504 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 24 points:

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

Polycyclic presentation:

2^11|b^2=i,d^2=i,b^a=bi,c^b=ci,d^a=di,d^c=dik,g^e=gk,g^f=gij,h^e=hj,h^f=hk

Conjugacy classes & character table

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