#290972

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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

804 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)(9,10,15,18)(11,16,21,22)(12,20,23,24)(13,17,14,19)(25,26,30,34)(27,35,38,39)(28,33,29,32)(31,36,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,20)(14,24)(15,21)(17,23)(18,22)(27,36)(28,29)(31,39)(32,33)(35,40)(37,38);(9,12,13,16,15,23,14,22)(10,11,19,24,18,21,17,20)(25,27,29,36,30,38,28,37)(26,31,33,39,34,40,32,35);(2,6)(5,7)(9,13,15,14)(10,17,18,19)(11,20,21,24)(12,16,23,22)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=hi,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,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, 198168);