#7622994

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #3573 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530190);