#493210

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

814 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)(9,10,13,18,14,19,28,34,15,20,29,35,30,36,31,40)(11,16,22,32,23,12,24,26,21,27,37,39,38,25,17,33);(2,4)(5,6,8,7)(9,11)(10,16)(12,19)(13,22)(14,23)(15,21)(17,31)(18,32)(20,27)(24,28)(25,36)(26,34)(29,37)(30,38)(33,40)(35,39);(5,7,8,6)(11,21)(12,25)(16,27)(17,24)(22,37)(23,38)(26,33)(32,39);(9,12,13,26,14,27,28,39,15,25,29,33,30,16,31,32)(10,17,18,11,19,22,34,23,20,24,35,21,36,37,40,38)

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,d^a=di,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, 400406);