#303115

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
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	C₂³×C₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 210311);