#493054

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1160 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)(3,4)(5,6,8,7)(9,10,15,21)(11,24,27,30)(12,18,17,37)(13,19,34,23)(14,25,35,22)(16,36,33,20)(26,38,40,31)(28,29,39,32);(2,4)(9,11,16,28,13,26,14,18,15,27,33,39,34,40,35,37)(10,17,22,38,23,29,20,24,21,12,25,31,19,32,36,30);(5,7,8,6)(9,12)(10,18)(11,25)(13,32)(14,30)(15,17)(16,31)(19,28)(20,40)(21,37)(22,27)(23,39)(24,35)(26,36)(29,34)(33,38);(9,13,15,34)(10,19,21,23)(11,26,27,40)(12,29,17,32)(14,33,35,16)(18,39,37,28)(20,22,36,25)(24,38,30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=hi,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400250);