#72335048

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 13611 conjugacy classes of subgroups; only the 17280 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^11|c^2=g,e^2=g,b^a=bk,c^b=ci,d^a=dij,d^c=dh,e^b=ej,e^d=ehij,f^c=fhj,f^e=fhj

Conjugacy classes & character table

332 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335056, #72336510, #72337359, …
maximal normal subgroup of	#72335056, #72336510, #72337359, …

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