#503333

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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 ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=ei,f^2=hi,g^2=h,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=ghi,h^b=hi,d^c=di

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, 410529);