#483490

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 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)(4,7)(6,8)(9,10,14,21,16,20,33,38,15,19,32,23,13,29,31,39)(11,22,25,28,24,17,34,27,18,36,37,26,30,35,40,12);(2,5)(3,8)(6,7)(9,11)(10,17)(12,23)(13,30)(14,25)(15,18)(16,24)(19,35)(20,36)(21,27)(22,29)(26,38)(28,39)(31,40)(32,37)(33,34);(9,12,14,22,16,28,33,17,15,27,32,36,13,26,31,35)(10,18,21,37,20,30,38,40,19,11,23,25,29,24,39,34);(10,19)(11,18)(20,29)(21,23)(24,30)(25,37)(34,40)(38,39)

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=di,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, 390686);