#72268588

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

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

2783332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^11|b^2=h,g^2=k,b^a=bj,d^b=dh,e^b=eh,g^b=gk,g^c=gj,g^e=gi,g^f=gh

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268601, C₂×#72268588
maximal normal subgroup of	#72268601, C₂×#72268588
maximal quotient of	#72287919, C₂×#72268588

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