#6569396

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=fghi,e^2=hi,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, 6476592);