#501263

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,f^2=i,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,f^c=fi,e^d=ehi,f^d=fi,h^d=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, 408459);