#411248

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

650 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)(3,8)(4,6)(5,7)(9,10,16,23,14,20,15,21)(11,17,25,33,24,13,19,31)(12,26,32,37,29,18,30,38)(22,28,27,40,39,34,36,35);(9,11,16,25,14,24,15,19)(10,17,21,31,20,13,23,33)(12,27,32,39,29,36,30,22)(18,35,37,34,26,40,38,28);(1,3,5,4)(2,6,7,8)(9,12,16,32,14,29,15,30)(10,18,23,38,20,26,21,37)(11,22,19,36,24,39,25,27)(13,28,33,35,17,34,31,40);(1,4,5,3)(2,6,7,8)(9,13,14,17)(10,19,20,25)(11,23,24,21)(12,28,29,34)(15,31,16,33)(18,36,26,27)(22,38,39,37)(30,40,32,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=fg,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^d=ei

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, 318444);