#295954

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 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)(9,10,15,18)(11,16,21,22)(12,20,23,24)(13,17,14,19)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(2,5)(6,7)(9,11,15,21)(10,16,18,22)(12,17,23,19)(13,20,14,24)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(9,12,13,16,15,23,14,22)(10,11,19,24,18,21,17,20)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,17,18,19)(11,20,21,24)(12,16,23,22)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=fh,d^2=ehi,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, 203150);