#506466

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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:

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

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,d^b=di,f^c=fhi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413662);