#330220

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×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 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)(9,10,13,17,12,16,21,24)(11,18,20,23,19,14,22,15)(25,26,28,32,29,33,37,31)(27,30,34,39,35,38,36,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,22)(23,24)(25,27,29,35)(26,30,33,38)(28,34,37,36)(31,40,32,39);(3,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,17)(16,23)(18,24)(20,21)(26,31)(27,34)(28,37)(32,33)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(11,19)(14,18)(15,23)(20,22)(27,35)(30,38)(34,36)(39,40)

Polycyclic presentation:

2^9|a^2=f,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, 237416);