#493126

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	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

824 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)(9,10,14,19,15,20,29,35,13,18,26,34,28,12,17,27)(11,21,24,37,25,22,36,38,23,16,30,32,39,33,31,40);(2,4)(5,6,7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,31)(18,21)(19,32)(20,33)(26,30)(27,38)(28,39)(29,36)(34,37)(35,40);(9,12,26,35,15,10,17,34,13,20,14,27,28,18,29,19)(11,22,30,40,25,16,31,37,23,33,24,38,39,21,36,32);(10,18)(11,23)(12,20)(19,34)(24,30)(25,39)(27,35)(31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=be,d^a=di,d^b=di,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, 400322);