#453877

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
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

545 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)(4,5)(7,8)(9,10,16,20,14,21,15,33)(11,17,19,32,23,31,24,13)(12,25,30,37,28,39,29,18)(22,35,36,40,38,27,26,34);(9,11,16,24,14,23,15,19)(10,17,20,13,21,31,33,32)(12,26,30,38,28,36,29,22)(18,35,25,34,37,27,39,40);(2,5)(6,8)(9,12,15,29,14,28,16,30)(10,18,20,25,21,37,33,39)(11,22,24,26,23,38,19,36)(13,27,17,34,32,35,31,40);(1,3,4,7)(2,6,5,8)(9,13,15,17,14,32,16,31)(10,19,20,11,21,24,33,23)(12,27,29,34,28,35,30,40)(18,36,25,22,37,26,39,38)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=g,d^2=f,f^2=i,g^2=i,h^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,f^b=fi,g^b=gi,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, 361073);