#501267

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

1051 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 408463);