#6675865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,31)(23,32,27,28)(24,35,38,43)(25,39,41,30)(29,36,45,33)(34,42,40,46)(37,50,51,48)(44,47,52,49);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,27)(22,28,31,32)(24,36,38,33)(25,40,41,34)(29,43,45,35)(30,46,39,42)(37,47,51,49)(44,48,52,50);(2,4)(21,24,26,38)(22,29,31,45)(23,33,27,36)(25,37,41,51)(28,35,32,43)(30,44,39,52)(34,47,40,49)(42,48,46,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,41)(22,30,31,39)(23,34,27,40)(24,37,38,51)(28,42,32,46)(29,44,45,52)(33,47,36,49)(35,48,43,50);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,31)(25,41)(28,32)(29,45)(34,40)(35,43)(37,51)(47,49)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583061);