#465399

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

865 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 372595);