#505653

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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)(4,5)(6,8)(9,10,16,19,14,20,13,27)(11,22,23,33,12,17,25,35)(15,30,32,39,29,40,28,21)(18,36,34,26,38,31,37,24);(2,5)(6,7)(9,11)(10,17)(12,14)(13,25)(15,26)(16,23)(18,30)(19,35)(20,22)(21,37)(24,29)(27,33)(28,36)(31,32)(34,39)(38,40);(9,12,14,11)(10,18,20,38)(13,25,16,23)(15,31,29,36)(17,30,22,40)(19,37,27,34)(21,33,39,35)(24,32,26,28);(1,3,4,8)(2,6,5,7)(9,13,14,16)(10,19,20,27)(11,23,12,25)(15,28,29,32)(17,33,22,35)(18,37,38,34)(21,30,39,40)(24,36,26,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,h^c=hi

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, 412849);