#7622524

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂³×D₄
commutator	C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂³×C₄⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,6,12,7,11,26,35,8,13,27,36,5,24,25,50)(3,14,18,39,19,30,45,54,16,38,44,59,17,9,28,31)(4,10,22,34,23,33,49,56,20,32,46,55,21,47,48,62)(15,37,42,58,43,52,61,63,40,57,60,64,41,29,51,53);(1,3,5,17,8,16,7,19)(2,9,11,14,13,30,24,38)(4,15,21,41,20,40,23,43)(6,18,25,28,27,44,26,45)(10,29,33,37,32,52,47,57)(12,31,35,39,36,54,50,59)(22,42,48,51,46,60,49,61)(34,53,56,58,55,63,62,64);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,29)(11,33)(12,34)(13,32)(14,37)(16,40)(17,41)(18,42)(19,43)(24,47)(25,48)(26,49)(27,46)(28,51)(30,52)(31,53)(35,56)(36,55)(38,57)(39,58)(44,60)(45,61)(50,62)(54,63)(59,64);(4,20)(10,32)(15,40)(21,23)(22,46)(29,52)(33,47)(34,55)(37,57)(41,43)(42,60)(48,49)(51,61)(53,63)(56,62)(58,64);(3,16)(9,30)(14,38)(15,40)(17,19)(18,44)(28,45)(29,52)(31,54)(37,57)(39,59)(41,43)(42,60)(51,61)(53,63)(58,64)

Polycyclic presentation:

2^9|a^2=g,b^2=f,f^2=i,g^2=h,h^2=i,b^a=bf,f^a=fi,e^b=ei,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529720);