#6702095

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,31)(21,24)(22,26)(25,33)(27,43)(28,45)(29,39)(30,37)(32,48)(34,44)(35,40)(36,42)(38,46)(41,47);(9,11,14,10)(12,15,13,16)(17,19,21,29)(18,23,24,39)(20,32,33,47)(22,27,35,30)(25,41,31,48)(26,43,40,37)(28,36,46,34)(38,44,45,42);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(18,24)(19,27)(22,35)(23,37)(25,31)(28,32)(29,30)(34,36)(38,48)(39,43)(41,45)(46,47);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(20,33)(25,31)(28,46)(32,47)(34,36)(38,45)(41,48)(42,44);(17,20,21,33)(18,25,24,31)(19,28,29,46)(22,36,35,34)(23,38,39,45)(26,44,40,42)(27,32,30,47)(37,48,43,41)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6609291);