#6561358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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

1347 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)(19,28,30,43)(20,33,38,44)(21,25,35,26)(23,31,40,32)(24,36,42,37)(29,41,34,39)(45,47,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,32,22,30,35,31)(18,23,26,28,27,40,25,43)(20,34,37,47,38,29,36,46)(24,41,33,48,42,39,44,45);(10,13)(14,16)(17,20,35,36,22,38,21,37)(18,24,25,44,27,42,26,33)(19,29,31,47,30,34,32,46)(23,39,43,48,40,41,28,45);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(23,40)(28,43)(29,34)(31,32)(39,41)(45,48)(46,47);(11,15)(14,16)(17,21,22,35)(18,25,27,26)(19,31,30,32)(20,36,38,37)(23,28,40,43)(24,33,42,44)(29,46,34,47)(39,48,41,45)

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^b=ei,e^c=ei,e^d=ehi

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, 6468554);