#483116

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)(9,10,14,20,12,18,27,19,15,17,30,36,28,35,13,29)(11,21,24,31,26,37,40,32,25,38,39,16,22,33,23,34);(2,5)(3,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,31)(18,32)(19,37)(20,38)(21,36)(26,28)(27,39)(29,33)(30,40)(34,35);(2,5)(6,8)(10,17)(16,31)(18,35)(19,29)(20,36)(21,38)(32,34)(33,37);(9,12,15,28)(10,18,17,35)(11,22,25,26)(13,14,27,30)(16,32,31,34)(19,36,29,20)(21,33,38,37)(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=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, 390312);