#466764

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1369 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)(9,10,15,20)(11,21,26,38)(12,27,29,39)(13,18,31,36)(14,32,22,19)(16,33,35,25)(17,24,37,28)(23,34,30,40);(1,2,3,4)(9,11,22,33,15,26,14,25)(10,16,32,38,20,35,19,21)(12,13,24,23,29,31,28,30)(17,36,39,34,37,18,27,40);(2,4)(5,6,8,7)(9,12,15,29)(10,17,20,37)(11,23,26,30)(13,25,31,33)(14,24,22,28)(16,34,35,40)(18,38,36,21)(19,39,32,27);(5,7,8,6)(9,13,22,23,15,31,14,30)(10,18,32,40,20,36,19,34)(11,24,33,29,26,28,25,12)(16,27,38,17,35,39,21,37)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 373960);