#500672

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	#60647	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1206 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,g^c=gi,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, 407868);