#483107

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
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

1079 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)(9,10,14,21,15,22,30,35,16,18,28,36,12,19,13,20)(11,23,26,32,24,37,25,38,27,39,29,17,31,34,40,33);(2,5)(3,7)(9,11)(10,17)(12,24)(13,29)(14,25)(15,31)(16,27)(18,32)(19,33)(20,37)(21,23)(22,38)(26,30)(28,40)(34,35)(36,39);(2,5)(6,8)(10,18)(17,32)(19,22)(20,35)(21,36)(23,39)(33,38)(34,37);(9,12,16,15)(10,19,18,22)(11,24,27,31)(13,28,30,14)(17,33,32,38)(20,36,35,21)(23,37,39,34)(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=cgi,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390303);