#465306

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

Polycyclic presentation:

2^9|b^2=egi,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, 372502);