#6561262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6468458);