#500754

order	512
nilpotency class	4
isoclinism	isoclinic class of #500753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407950);