#6561210

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(19,28,30,42)(20,33,34,41)(21,25,36,26)(23,31,38,32)(24,39,40,35)(29,44,47,37)(43,45,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,32,22,30,36,31)(18,23,26,28,27,38,25,42)(20,29,35,46,34,47,39,45)(24,37,33,43,40,44,41,48);(10,13)(14,16)(17,20,21,35,22,34,36,39)(18,24,26,33,27,40,25,41)(19,29,32,46,30,47,31,45)(23,37,28,43,38,44,42,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,34)(23,38)(24,40)(28,42)(31,32)(33,41)(35,39);(11,15)(14,16)(17,21,22,36)(18,25,27,26)(19,31,30,32)(20,35,34,39)(23,28,38,42)(24,41,40,33)(29,45,47,46)(37,43,44,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,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^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, 6468406);