#290116

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

520 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)(9,10,14,23,12,17,13,18)(11,19,22,16,20,15,21,24)(25,26,30,32,28,33,29,34)(27,35,38,39,36,31,37,40);(1,3,5,4)(2,6,8,7)(9,11,14,22,12,20,13,21)(10,15,23,24,17,19,18,16)(25,27,29,37,28,36,30,38)(26,31,34,39,33,35,32,40);(9,11,12,20)(10,16,17,24)(13,22,14,21)(15,23,19,18)(26,32)(27,36)(29,30)(31,39)(33,34)(35,40);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,24)(18,23)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,c^2=hi,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, 197312);