#501259

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61030	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1084 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,e^d=ehi,f^d=fi,h^d=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, 408455);