#499316

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=i,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,d^b=di,e^c=eh,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406512);