#484090

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391286);