#72320685

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	D₈×#41	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=e,c^2=efhj,d^2=efg,e^2=hi,f^2=h,g^2=ij,h^2=j,c^b=cf,d^a=dg,d^b=dhj,d^c=dhij,e^c=eh,e^d=ej,f^b=fj,f^c=fh,f^d=fj,g^a=gij,h^c=hj

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320730, #72321057
maximal normal subgroup of	#72320730, #72321057

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