#500926

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408122);