#72322682

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂³×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
socle	C₂⁴	C₂²×#97
prime radical	C₂³×D₄	#97
LCS	G ▷ C₂³×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5438 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)(17,18,19,24,20,25,27,32,21,23,26,31,28,22,29,30);(2,4)(9,10,11,14,12,13,15,16)(18,22,23,25)(19,26)(24,32,31,30)(27,29);(2,4);(1,2,3,4)(6,8)(10,13)(14,16)

Polycyclic presentation:

2^11|a^2=h,b^2=e,d^2=fi,e^2=i,h^2=j,j^2=k,b^a=bj,d^a=dg,d^b=df,d^c=dfi,e^a=ek,h^b=hk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322777, #72322912
maximal normal subgroup of	#72322777, #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.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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);