#509205

order	512
nilpotency class	4
isoclinism	isoclinic class of #509203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30499
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

737 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:

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

Polycyclic presentation:

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

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, 416401);