#500929

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

739 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)(11,20)(12,25)(13,31)(14,19)(15,22)(16,29)(17,35)(18,28)(21,24)(23,37)(26,30)(27,38)(32,36)(33,40)(34,39);(1,3,5,4)(2,6,8,7)(9,10)(11,21)(12,26)(13,31)(14,19)(15,22)(16,35)(17,29)(18,28)(20,24)(23,36)(25,30)(27,38)(32,37)(33,40)(34,39);(9,11,14,24)(10,15,19,34)(12,27,30,40)(13,32,28,23)(16,36,17,37)(18,38,31,33)(20,26,21,25)(22,35,39,29);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=e,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, 408125);