#506194

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413390);