#287319

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194515);