#500925

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

717 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)(9,10,13,20)(11,23,26,17)(12,22,16,19)(14,18,29,38)(15,32,30,21)(24,37,27,40)(25,35,33,34)(28,36,31,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,25)(13,26)(14,31)(15,24)(16,33)(18,36)(19,35)(20,23)(21,37)(22,34)(27,30)(28,29)(32,40)(38,39);(10,18)(11,24)(12,16)(14,30)(15,29)(17,34)(19,32)(20,38)(21,22)(23,35)(25,31)(26,27)(28,33)(37,40);(1,4,5,3)(2,7,8,6)(9,12,13,16)(10,19,20,22)(11,25,26,33)(14,15,29,30)(17,35,23,34)(18,21,38,32)(24,28,27,31)(36,37,39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,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, 408121);