#317663

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
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

524 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ghi,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224859);