#500931

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ 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)(12,17)(13,27)(14,32)(15,20)(16,24)(18,30)(19,26)(22,25)(23,38)(28,31)(29,34)(33,37)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,10)(11,22)(12,17)(13,28)(14,32)(15,20)(16,24)(18,26)(19,30)(21,25)(23,37)(27,31)(29,34)(33,38)(35,40)(36,39);(9,11,15,25)(10,16,20,36)(12,23,14,33)(13,29,31,40)(17,34,32,35)(18,37,19,38)(21,28,22,27)(24,26,39,30);(1,4,5,3)(2,7,8,6)(9,12,15,14)(10,17,20,32)(11,23,25,33)(13,26,31,30)(16,34,36,35)(18,27,19,28)(21,38,22,37)(24,29,39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,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, 408127);