#500890

order	512
nilpotency class	4
isoclinism	isoclinic class of #500887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408086);