#324851

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
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

490 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)(9,10,12,15)(11,17,18,24)(13,21,20,16)(14,22,23,19)(25,26,28,31)(27,33,36,30)(29,32,34,38)(35,40,37,39);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,21)(22,24)(25,27,34,35,28,36,29,37)(26,30,38,39,31,33,32,40);(9,11,13,19,12,18,20,22)(10,14,16,17,15,23,21,24)(27,35)(29,34)(30,39)(32,38)(33,40)(36,37);(1,3,4,8)(2,6,5,7)(10,15)(14,23)(16,21)(17,24)(26,31)(30,33)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 232047);