#501667

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(9,10,17,32,13,28,31,40,14,21,15,22,16,20,29,34)(11,23,27,36,26,37,33,38,12,18,25,39,24,35,30,19);(1,2,4,3)(5,6)(7,8)(9,11,14,12)(10,18,21,23)(13,26,16,24)(15,25,17,27)(19,34,38,40)(20,37,28,35)(22,36,32,39)(29,30,31,33);(6,8)(9,12,14,11)(10,19,21,38)(13,26,16,24)(15,30,17,33)(18,34,23,40)(20,36,28,39)(22,37,32,35)(25,31,27,29);(1,3,4,2)(9,13,14,16)(10,20,21,28)(11,24,12,26)(15,29,17,31)(18,35,23,37)(19,36,38,39)(22,40,32,34)(25,33,27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408863);