#501018

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1027 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,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, 408214);