#501655

order	512
nilpotency class	4
isoclinism	isoclinic class of #501655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33605	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,e^2=i,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^c=fh,h^c=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, 408851);