#492938

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×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×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1340 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);(2,4)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(5,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,g^2=h,h^2=i,b^a=be,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, 400134);