#501620

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 408816);