#409526

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 316722);