#501403

order	512
nilpotency class	4
isoclinism	isoclinic class of #501402
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32939	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=eh,f^2=h,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, 408599);