#72322585

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×#106
commutator	C₂²×C₈	C₂³×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂³×D₄	#106
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄²×D₄ ◁ C₂×C₄×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4908 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)(17,18,19,22,20,24,25,29,21,23,26,28,27,30,32,31);(9,10,12,15,13,16,11,14)(18,22,23,28)(19,25)(20,27)(24,31,30,29)(26,32);(2,4);(1,2,3,4)(6,8)(9,11,13,12)(10,14,16,15)

Polycyclic presentation:

2^11|a^2=h,b^2=e,d^2=fi,e^2=i,h^2=j,j^2=k,b^a=bhjk,d^a=dg,d^c=df,e^a=ek,h^b=hj,j^b=jk

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322777, #72322866
maximal normal subgroup of	#72322777, #72322866

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