#462840

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 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)(3,6)(4,8)(5,7)(9,10,13,16,15,20,14,19)(11,21,23,31,25,17,24,33)(12,26,28,18,29,32,30,36)(22,38,27,37,39,35,40,34);(1,3,5,4)(2,6,7,8)(10,16,20,19)(17,33,21,31)(18,32,36,26)(34,38,37,35);(2,7)(6,8)(9,11,15,25)(10,17,20,21)(12,27,29,40)(13,24,14,23)(16,31,19,33)(18,35,36,38)(22,30,39,28)(26,34,32,37);(1,4,5,3)(2,8,7,6)(9,12,13,28,15,29,14,30)(10,18,16,32,20,36,19,26)(11,22,23,27,25,39,24,40)(17,34,33,38,21,37,31,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=ef,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,d^c=de,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370036);