#72233698

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

At least 12124 conjugacy classes of subgroups; only the 6600 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^11|d^2=k,e^2=k,b^a=bj,d^c=dh,e^c=eg,e^d=ek,f^c=fi,f^d=fj,g^d=gk,g^f=gj,h^e=hk,i^e=ij

Conjugacy classes & character table

206 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233730, #72258471, #72258990, …
maximal normal subgroup of	#72258471, #72258990, #72260825, …

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