#330221

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

619 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)(3,6)(4,5)(7,8)(9,10,14,23,12,17,13,18)(11,19,22,16,20,15,21,24)(25,26,30,32,29,34,28,33)(27,31,37,40,36,38,35,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,19)(23,24)(25,27,29,36)(26,31,34,38)(28,35,30,37)(32,40,33,39);(3,7)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,18)(17,24)(19,23)(26,32)(27,35)(28,30)(33,34)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(11,20)(15,19)(16,24)(21,22)(27,36)(31,38)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=ehi,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, 237417);