#483525

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
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	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 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:

(2,5)(3,7)(6,8)(9,10,14,20,13,21,29,38,15,18,31,23,30,19,36,37)(11,22,25,27,26,33,39,28,17,34,35,40,24,16,32,12);(1,2,4,6)(3,8,7,5)(9,11,15,17)(10,16,18,33)(12,23,28,20)(13,26,30,24)(14,25,31,35)(19,34,21,22)(27,37,40,38)(29,39,36,32);(9,12,14,22,13,27,29,33,15,28,31,34,30,40,36,16)(10,17,20,35,21,24,38,32,18,11,23,25,19,26,37,39);(10,18)(11,17)(19,21)(20,23)(24,26)(25,35)(32,39)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,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, 390721);