#417292

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

732 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,27)(13,19)(14,20)(15,29)(16,21)(18,30)(22,38)(23,34)(24,40)(25,35)(26,36)(28,33)(31,32)(37,39);(1,3,4,7)(2,5,6,8)(9,11,15,24,14,26,16,25)(10,17,21,35,20,36,29,40)(12,28,30,39,19,34,32,22)(13,23,18,37,27,33,31,38);(3,7)(5,8)(9,12,14,19)(10,18,20,31)(11,22)(13,29,27,21)(15,32,16,30)(17,33)(23,36)(24,34)(25,28)(26,39)(35,37)(38,40);(2,6)(5,8)(9,13,16,31,14,27,15,18)(10,19,21,32,20,12,29,30)(11,23,25,38,26,33,24,37)(17,34,35,22,36,28,40,39)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324488);