#480155

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=i,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,c^b=ci,e^b=ei,f^b=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387351);