#483057

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

723 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)(9,10,14,19,17,24,30,34,15,20,13,29,16,11,21,23)(12,18,27,33,31,36,39,40,25,32,26,38,28,22,35,37);(1,2,4,6)(3,8,7,5)(9,11,15,24)(10,17,20,16)(12,22,25,36)(13,29,14,19)(18,31,32,28)(21,34,30,23)(26,38,27,33)(35,40,39,37);(9,12)(10,18)(11,22)(13,26)(14,27)(15,25)(16,28)(17,31)(19,33)(20,32)(21,35)(23,37)(24,36)(29,38)(30,39)(34,40);(12,25)(18,32)(22,36)(26,27)(28,31)(33,38)(35,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=gh,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, 390253);