#452756

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,h^d=hi

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, 359952);