#6558816

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

1305 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)(17,18,23,27)(19,28,31,39)(20,32,33,46)(21,26,22,37)(24,36,40,30)(25,34,41,35)(29,43,44,38)(42,47,48,45);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,36,23,31,22,30)(18,24,37,39,27,40,26,28)(20,29,34,47,33,44,35,45)(25,38,46,48,41,43,32,42);(10,13)(14,16)(17,20,21,34,23,33,22,35)(18,25,37,46,27,41,26,32)(19,29,36,47,31,44,30,45)(24,38,39,48,40,43,28,42);(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|a^2=i,b^2=fi,c^2=gi,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, 6466012);