#438978

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346174);