#483624

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×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×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

458 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)(9,10,14,19,15,20,29,36,13,18,27,34,28,35,31,40)(11,21,23,37,24,25,33,39,22,16,30,32,38,12,17,26);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,24)(17,31)(18,21)(19,32)(25,35)(26,36)(27,30)(28,38)(29,33)(34,37)(39,40);(9,12,14,26,15,21,29,37,13,25,27,39,28,16,31,32)(10,17,19,11,20,23,36,24,18,33,34,22,35,30,40,38);(2,6)(5,7)(10,18)(11,22)(17,33)(19,34)(20,35)(23,30)(24,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390820);