#483632

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3171	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 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)(3,5)(4,6)(7,8)(9,10,13,19,14,20,28,36,15,18,29,34,30,35,31,40)(11,16,22,32,23,12,24,25,21,26,37,39,38,27,17,33);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,31)(18,26)(19,32)(24,28)(25,36)(27,35)(29,37)(30,38)(33,40)(34,39);(9,12,13,25,14,26,28,39,15,27,29,33,30,16,31,32)(10,17,19,11,20,22,36,23,18,24,34,21,35,37,40,38);(2,6)(5,7)(10,18)(11,21)(17,24)(19,34)(20,35)(22,37)(23,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390828);