#505651

order	512
nilpotency class	4
isoclinism	isoclinic class of #500109
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×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

479 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)(9,10,15,18,13,19,14,28)(11,20,29,38,21,37,23,35)(12,24,30,34,25,40,27,17)(16,31,32,22,33,26,36,39);(2,5)(6,7)(11,21)(12,25)(17,34)(20,37)(23,29)(24,40)(27,30)(35,38);(9,11,13,21)(10,16,19,33)(12,26,25,31)(14,29,15,23)(17,35,34,38)(18,36,28,32)(20,24,37,40)(22,27,39,30);(1,3,4,8)(2,6,5,7)

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,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, 412847);