#480085

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1072 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)(3,7,8,5)(4,6)(9,10,14,20,15,21,27,34,16,22,12,18,13,19,28,35)(11,23,24,38,26,33,39,37,25,36,30,40,29,17,31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,29)(14,30)(15,26)(16,25)(18,32)(19,36)(20,37)(21,23)(22,33)(27,31)(28,39)(34,40)(35,38);(9,12,15,28,16,14,13,27)(10,18,21,35,22,20,19,34)(11,24,26,39,25,30,29,31)(17,32,23,38,33,37,36,40);(11,25)(17,33)(23,36)(24,30)(26,29)(31,39)(32,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387281);