#500930

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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

759 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)(9,10,12,18)(11,21,23,16)(13,17,14,29)(15,31,26,20)(19,30,36,28)(22,34,27,33)(24,37,39,40)(25,35,32,38);(1,3,5,4)(2,6,8,7)(9,11,12,23)(10,16,18,21)(13,27,14,22)(15,32,26,25)(17,34,29,33)(19,37,36,40)(20,38,31,35)(24,28,39,30);(10,17)(11,22)(13,28)(14,30)(15,26)(18,29)(19,20)(23,27)(24,25)(31,36)(32,39)(33,40)(34,37)(35,38);(1,4,5,3)(2,7,8,6)(10,18)(16,21)(17,29)(19,36)(20,31)(33,34)(35,38)(37,40)

Polycyclic presentation:

2^9|a^2=e,b^2=e,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, 408126);