#290115

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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

528 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)(9,10,14,16,12,17,13,18)(11,19,20,23,21,15,22,24)(25,26,28,31)(27,33,35,30)(29,32,36,39)(34,40,37,38);(1,3,5,4)(2,6,8,7)(9,11,14,20,12,21,13,22)(10,15,16,24,17,19,18,23)(27,34)(29,36)(30,38)(32,39)(33,40)(35,37);(10,16)(11,20)(13,14)(15,19)(17,18)(21,22)(25,27,28,35)(26,30,31,33)(29,37,36,34)(32,40,39,38);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,18)(23,24)

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,c^2=i,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197311);