#480156

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

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×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

971 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)(9,10,14,20,16,19,28,36,15,21,27,37,13,29,12,18)(11,22,26,32,25,17,31,34,24,35,39,40,30,33,23,38);(2,5)(3,7)(9,11)(10,17)(12,23)(13,30)(14,26)(15,24)(16,25)(18,32)(19,35)(20,34)(21,33)(22,29)(27,39)(28,31)(36,40)(37,38);(9,12,13,27,15,28,16,14)(10,18,29,37,21,36,19,20)(11,23,30,39,24,31,25,26)(17,32,22,38,33,40,35,34);(11,24)(17,33)(22,35)(23,31)(25,30)(26,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,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, 387352);