#480121

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387317);