#287162

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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)(4,5)(6,8)(9,10,13,18,15,20,14,19)(11,16,22,17,12,23,21,24)(25,26,30,32,29,33,28,34)(27,35,37,39,36,31,38,40);(2,5)(6,7)(9,11,13,22,15,12,14,21)(10,16,18,17,20,23,19,24)(25,27,30,37,29,36,28,38)(26,31,32,40,33,35,34,39);(9,12,15,11)(10,17,20,24)(13,22,14,21)(16,19,23,18)(26,32)(27,36)(28,30)(31,39)(33,34)(35,40);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,20,19)(11,21,12,22)(16,24,23,17)(25,28,29,30)(26,32,33,34)(27,37,36,38)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194358);