#483120

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=ghi,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, 390316);