#72298586

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

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

At least 23124 conjugacy classes of subgroups; only the 6300 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|b^2=h,d^2=i,e^2=h,b^a=bk,c^b=chi,d^a=dhij,d^b=dhi,d^c=dhi,e^a=eh,e^c=ehj,e^d=ehi,f^b=fg,f^c=fgi,f^d=fi,f^e=fg

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72298631
maximal normal subgroup of	#72298631

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