#438498

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

904 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345694);