#7622565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370843
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529761);