#503434

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#61084	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

977 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)(11,21)(12,16)(13,29)(14,18)(15,31)(17,25)(19,27)(20,28)(22,36)(23,33)(24,38)(26,34)(30,37)(32,39)(35,40);(1,3,5,4)(2,6,8,7)(10,16)(11,22)(13,14)(15,28)(17,34)(18,19)(21,37)(23,27)(24,30)(25,40)(26,39)(29,33)(32,35)(36,38);(1,3,5,4)(2,7,8,6)(9,11,23,22,15,26,13,25,12,24,14,30,28,40,27,39)(10,17,19,34,20,36,18,21,16,32,29,35,31,37,33,38);(1,4,5,3)(2,7,8,6)(10,16)(17,32)(18,33)(19,29)(20,31)(21,38)(34,35)(36,37)

Polycyclic presentation:

2^9|b^2=ei,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=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410630);