#294789

order	512
nilpotency class	3
isoclinism	isoclinic class of #271003
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 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)(4,5)(6,8)(9,10)(11,18)(12,20)(13,17)(14,15)(16,22)(19,24)(21,23)(25,26)(27,32)(28,37)(29,33)(30,38)(31,34)(35,39)(36,40);(2,5)(6,7)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,23,24,20)(25,27,31,36)(26,32,34,40)(28,29,35,30)(33,39,38,37);(9,12,14,22,13,21,11,19)(10,16,18,20,17,24,15,23)(25,28,30,27,31,35,29,36)(26,32,33,37,34,40,38,39);(1,3,4,8)(2,6,5,7)(10,17)(15,18)(16,24)(20,23)(25,29,31,30)(26,33,34,38)(27,35,36,28)(32,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201985);