#503443

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61090	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

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, 410639);