#252208

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

878 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,29,32)(27,30,34,38)(28,31,35,39)(33,37,36,40);(1,3,4,8)(2,5,6,7)(10,14)(13,20)(17,24)(19,22)(27,33)(28,35)(30,37)(31,39)(34,36)(38,40);(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,27,29,34)(26,30,32,38)(28,36,35,33)(31,40,39,37);(2,6)(5,7)(10,14)(11,18)(16,21)(19,22)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 159404);