#483104

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1089 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)(9,10,14,20,15,21,30,37,16,22,28,35,12,18,13,19)(11,23,26,34,24,36,25,38,27,39,29,17,31,33,40,32);(2,5)(3,7)(9,11)(10,17)(12,24)(13,29)(14,25)(15,31)(16,27)(18,32)(19,36)(20,23)(21,38)(22,34)(26,30)(28,40)(33,37)(35,39);(2,5)(6,8);(9,12,16,15)(10,18,22,21)(11,24,27,31)(13,28,30,14)(17,32,34,38)(19,35,37,20)(23,36,39,33)(25,29,40,26)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

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, 390300);