#72225883

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#392111
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂³×C₂²⋊C₄	C₂⁴
prime radical	C₂×#7670203	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×#35390 ◁ G
LECS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

125138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^11|a^2=e,c^2=i,e^2=ijk,b^a=bf,c^a=cg,c^b=ci,d^a=dh,d^b=dk,d^c=dk,e^b=ei,e^c=ej,e^d=ejk,f^a=fi,f^e=fk,g^a=gj,g^b=gk,g^c=gk,g^e=gk,h^a=hjk,h^e=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225910, #72226177, #72226360, …
maximal normal subgroup of	#72225910, #72226177, #72226360, …

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