#72319682

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×#21503
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72319626	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #180817 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,11,15)(4,14,12,7)(9,10,16,13)(17,18,24,29)(19,25,32,41)(20,26,31,40)(21,34,35,47)(22,28,23,38)(27,36,43,37)(30,44,45,42)(33,39,46,48);(2,6,8,15)(3,10)(4,12)(7,9,14,16)(11,13)(17,19,22,31,24,32,23,20)(18,25,28,40,29,41,38,26)(21,30,36,46,35,45,37,33)(27,39,34,44,43,48,47,42);(2,7,8,14)(3,11)(6,9,15,16)(10,13)(17,20,23,32,24,31,22,19)(18,26,38,41,29,40,28,25)(21,33,37,45,35,46,36,30)(27,42,47,48,43,44,34,39);(21,35)(27,43)(30,45)(33,46)(34,47)(36,37)(39,48)(42,44);(17,21,23,37,24,35,22,36)(18,27,28,34,29,43,38,47)(19,30,20,33,32,45,31,46)(25,39,40,44,41,48,26,42)

Polycyclic presentation:

2^11|a^2=ik,b^2=f,c^2=fi,e^2=g,f^2=i,g^2=i,h^2=jk,j^2=k,b^a=bh,c^a=cjk,e^a=eg,e^d=ei,f^a=fk,g^a=gi,h^a=hj,h^b=hjk,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

automorphism group of

#13710

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