#72309663

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#21526
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72309639	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×D₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=k,c^2=fjk,d^2=k,e^2=g,g^2=i,h^2=j,i^2=k,b^a=bh,d^a=dj,e^a=eg,e^c=ei,f^e=fk,g^a=gi,g^c=gk,h^a=hj,h^b=hj,i^a=ik

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

automorphism group of

#12463

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