#291143

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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

820 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)(15,23,22,19)(25,26)(27,31)(28,33)(29,32)(30,34)(35,40)(36,38)(37,39);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(13,20)(15,22)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,38)(34,40);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)(26,32)(27,35)(28,30)(31,38)(36,37)(39,40);(2,6)(5,7)(10,14)(15,22)(16,21)(17,24)(25,28,29,30)(26,33,32,34)(27,36,37,35)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^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=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198339);