#72322448

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#442
commutator	C₂³×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂×#13325	C₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ C₂×#8260 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6)(7,8)(9,10,12,14,13,15,11,16)(17,18,20,24,21,26,29,27)(19,23,30,25,31,22,28,32);(1,2)(3,4)(18,22)(19,28)(20,29)(23,26)(24,25)(27,32)(30,31);(1,3,4,2)(6,8)(9,11,13,12)(10,14,15,16);(2,3)(6,8)(9,10,11,16,13,15,12,14)(17,19,29,28,21,31,20,30)(18,23,27,32,26,22,24,25)

Polycyclic presentation:

2^11|a^2=e,c^2=fik,d^2=ei,e^2=i,h^2=j,j^2=k,b^a=bh,c^a=cg,c^b=cf,d^a=dg,d^b=dfjk,d^c=dfik,e^b=ek,h^a=hj,h^b=hjk,h^d=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322727, #72322912
maximal normal subgroup of	#72322727, #72322912

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