#501257

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1109 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 408453);