#453806

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
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

567 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=ef,d^2=e,e^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,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 361002);