#290378

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197574);