#472912

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=i,d^2=ef,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,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380108);