#6540587

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,38)(30,39)(31,41)(33,40)(35,42)(36,44)(37,43)(45,46)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,41,26,40)(20,35,37,46,36,30,38,47)(25,42,44,48,43,39,34,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,37,23,36,22,38)(18,25,27,44,28,43,26,34)(19,30,32,47,33,35,31,46)(24,39,29,45,41,42,40,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(20,36)(25,43)(30,35)(34,44)(37,38)(39,42)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,36,38)(24,29,41,40)(25,34,43,44)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fg,e^2=i,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=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, 6447783);