#465208

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10)(11,17)(12,14)(13,28)(15,19)(16,20)(18,29)(21,25)(22,36)(23,26)(24,33)(27,31)(30,34)(32,37)(35,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11,15,24,16,25,14,23)(10,17,19,33,20,21,12,26)(13,22,30,38,31,39,29,37)(18,32,28,36,34,35,27,40);(3,7)(6,8)(9,12,15,20,16,19,14,10)(11,21,24,33,25,17,23,26)(13,27,30,34,31,28,29,18)(22,35,38,36,39,32,37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,27)(14,29)(15,30)(16,31)(17,32)(19,28)(20,34)(21,35)(23,37)(24,38)(25,39)(26,40)(33,36)

Polycyclic presentation:

2^9|b^2=eg,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372404);