#316472

order	512
nilpotency class	3
isoclinism	isoclinic class of #262787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 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)(4,6)(5,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,29,33)(27,35,38,30)(28,34,36,32)(31,37,40,39);(1,3,4,8)(2,5,6,7)(25,27,36,37,29,38,28,39)(26,30,32,31,33,35,34,40);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,27,29,38)(26,31,33,40)(28,37,36,39)(30,34,35,32);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,28,29,36)(26,32,33,34)(27,37,38,39)(30,40,35,31)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=di,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, 223668);