#466758

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1377 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373954);