#394374

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2641 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,g^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,e^c=ei,g^c=gi,h^d=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, 301570);