#290922

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

710 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)(11,16)(12,14)(13,17)(15,18)(19,22)(20,24)(21,23)(25,26)(27,32)(28,38)(29,34)(30,36)(31,35)(33,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,15,22)(10,16,18,19)(12,23,17,24)(13,20,14,21)(25,27,31,33)(26,32,35,39)(28,34,40,36)(29,37,30,38);(9,12,13,18,15,17,14,10)(11,19,20,24,22,16,21,23)(25,28,30,39,31,40,29,32)(26,33,34,38,35,27,36,37);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,20,22,21)(16,24,19,23)(25,29,31,30)(26,34,35,36)(27,37,33,38)(28,32,40,39)

Polycyclic presentation:

2^9|b^2=eh,c^2=fhi,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198118);