#330777

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

545 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=e,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237973);