#493223

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

828 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)(9,10,13,18,14,19,26,32,15,12,17,24,27,25,31,37)(11,16,22,29,23,30,36,39,20,21,28,34,33,35,38,40);(2,4)(5,6,8,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,28)(18,29)(19,30)(24,34)(25,35)(26,36)(27,33)(31,38)(32,39)(37,40);(5,7,8,6)(10,12)(11,20)(18,24)(19,25)(22,28)(23,33)(32,37)(36,38);(9,12,13,24,14,25,26,37,15,10,17,18,27,19,31,32)(11,21,22,34,23,35,36,40,20,16,28,29,33,30,38,39)

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^a=ci,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, 400419);