#72264999

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

410992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^11|b^a=bg,d^c=dh,e^a=ei,e^c=ej,f^b=fj,f^d=fk

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265090, #72333354
maximal normal subgroup of	#72265090, #72333354
maximal quotient of	#72265836, #72265895

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