#501061

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
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	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1073 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:

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

Polycyclic presentation:

2^9|a^2=fi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi,f^d=fi

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, 408257);