#499418

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2085
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406614);