#72258160

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₄×#102
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle	C₂³	#8202
prime radical	C₂²×D₈	#97
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#346 ◁ C₈×#346 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 40 points:

(26,33)(27,34)(28,32)(29,30)(35,39)(36,38)(37,40);(1,2,4,6,5,7,8,3)(10,14,16,22)(11,18)(15,20)(17,24,19,23)(25,26,28,34,31,35,32,36)(27,37,33,30,38,29,39,40);(1,3,8,7,5,6,4,2)(9,10,15,23,21,22,18,17,13,16,20,24,12,14,11,19)(25,27,29,34,28,39,40,26,31,38,37,36,32,33,30,35)

Polycyclic presentation:

2^11|b^2=d,c^2=dehj,d^2=g,e^2=h,f^2=i,h^2=j,i^2=k,b^a=bf,c^a=cfi,c^b=cfh,d^a=dk,d^c=dj,e^a=ei,e^b=eij,e^c=ej,f^a=fik,f^b=fi,f^c=fk,h^a=hk,h^b=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258190
maximal normal subgroup of	#72258190
maximal quotient of	#72258161

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