#438353

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1168 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)(9,10,16,21)(11,22,25,34)(12,26,31,36)(13,32,24,17)(14,20,15,19)(18,29,37,30)(23,33,38,39)(27,40,28,35);(5,6,7,8)(9,11,16,25)(10,17,21,32)(12,27,31,28)(13,15,24,14)(18,35,37,40)(19,34,20,22)(23,30,38,29)(26,39,36,33);(1,2,3,4)(6,8)(9,12,14,29,16,31,15,30)(10,18,19,36,21,37,20,26)(11,23,24,27,25,38,13,28)(17,33,22,35,32,39,34,40);(9,13,14,11,16,24,15,25)(10,17,19,22,21,32,20,34)(12,28,29,23,31,27,30,38)(18,33,36,35,37,39,26,40)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=fi,d^2=e,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^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=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, 345549);