#7622541

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂³×#107
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1845 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:

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

Polycyclic presentation:

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

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, 7529737);