#493208

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 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)(9,10,14,19,15,20,29,36,13,18,27,34,28,35,31,40)(11,21,23,37,24,25,33,39,22,16,30,32,38,12,17,26);(2,4)(5,6,8,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,24)(17,31)(18,21)(19,32)(25,35)(26,36)(27,30)(28,38)(29,33)(34,37)(39,40);(5,7,8,6)(11,22)(12,25)(16,21)(17,33)(23,30)(24,38)(26,39)(32,37);(9,12,14,26,15,21,29,37,13,25,27,39,28,16,31,32)(10,17,19,11,20,23,36,24,18,33,34,22,35,30,40,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^b=ci,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400404);