#6558810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 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)(3,7)(4,6)(5,8)(17,18)(19,28)(20,32)(21,26)(22,37)(23,27)(24,30)(25,35)(29,43)(31,39)(33,46)(34,41)(36,40)(38,44)(42,45)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30,23,31,21,36)(18,24,26,28,27,40,37,39)(20,29,35,45,33,44,34,47)(25,38,32,42,41,43,46,48);(10,13)(14,16)(17,20,22,35,23,33,21,34)(18,25,26,32,27,41,37,46)(19,29,30,45,31,44,36,47)(24,38,28,42,40,43,39,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,33)(25,41)(29,44)(32,46)(34,35)(38,43)(42,48)(45,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,30,31,36)(20,34,33,35)(24,39,40,28)(25,32,41,46)(29,45,44,47)(38,48,43,42)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466006);