#506464

order	512
nilpotency class	4
isoclinism	isoclinic class of #500887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,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, 413660);