#409674

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 316870);