#480039

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1082 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fg,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, 387235);