#72258182

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#442
commutator	C₄×C₈	C₂²×C₄²
Frattini, cosocle	C₂×C₈²	C₂⁴
socle	C₂²	#183994
prime radical	C₂×#509	M₁₆
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ #126544 ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^11|c^2=e,d^2=efh,e^2=h,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bg,c^a=cg,c^b=cgi,d^a=di,d^c=dhj,e^a=ei,e^b=eik,e^d=ej,f^a=fik,f^b=fik,f^c=fj,f^d=fj,g^a=gi,g^b=gi,g^c=gk,g^d=gk,h^a=hk,h^b=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72258190

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