#465305

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

762 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)(3,4)(9,10)(11,21)(12,16)(13,27)(14,19)(15,20)(17,23)(18,30)(22,36)(24,26)(25,33)(28,29)(31,34)(32,39)(35,38)(37,40);(1,2,5,6)(3,7,4,8)(9,11,16,25,14,23,15,24)(10,17,12,26,19,21,20,33)(13,22,31,39,29,37,30,38)(18,32,27,40,34,35,28,36);(1,3)(2,7)(4,5)(6,8)(9,12,16,10,14,20,15,19)(11,21,25,26,23,17,24,33)(13,28,31,34,29,27,30,18)(22,32,39,36,37,35,38,40);(1,4,5,3)(2,8,6,7)(9,13,14,29)(10,18,19,34)(11,22,23,37)(12,27,20,28)(15,30,16,31)(17,32,21,35)(24,38,25,39)(26,40,33,36)

Polycyclic presentation:

2^9|b^2=eg,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^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372501);