#481087

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

783 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)(9,10,13,19,15,16,26,29,14,18,27,32,11,20,22,33)(12,17,24,31,25,28,37,38,23,30,36,39,21,34,35,40);(1,2,4,6)(3,8,7,5)(9,11,14,15)(10,16,18,20)(12,21,23,25)(13,26,27,22)(17,28,30,34)(19,33,32,29)(24,37,36,35)(31,40,39,38);(9,12)(10,17)(11,21)(13,24)(14,23)(15,25)(16,28)(18,30)(19,31)(20,34)(22,35)(26,37)(27,36)(29,38)(32,39)(33,40);(12,23)(17,30)(21,25)(24,36)(28,34)(31,39)(35,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388283);