#483066

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

741 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)(9,10,13,16,15,20,12,25,14,19,26,30,27,34,28,18)(11,17,23,29,24,33,22,38,21,31,36,39,37,40,35,32);(1,2,4,6)(3,8,7,5)(10,16)(12,26)(13,28)(15,27)(17,29)(18,20)(19,30)(22,36)(23,35)(24,37)(25,34)(31,39)(32,33)(38,40);(9,11)(10,17)(12,22)(13,23)(14,21)(15,24)(16,29)(18,32)(19,31)(20,33)(25,38)(26,36)(27,37)(28,35)(30,39)(34,40);(11,21)(17,31)(22,35)(23,36)(24,37)(29,39)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

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, 390262);