#506413

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
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	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,19)(7,21,26,42)(8,28,32,62)(9,18,34,49)(10,20,13,17)(11,16,35,48)(14,27,41,24)(15,43,46,30)(22,54,29,63)(23,47,33,39)(25,40,58,53)(31,64,37,50)(36,51,65,45)(38,66,44,59)(52,55,68,57)(56,67,60,61);(1,2,3,4)(5,7,13,27,12,26,10,24)(6,14,20,42,19,41,17,21)(8,29,37,56,32,22,31,60)(9,33,35,58,34,23,11,25)(15,44,51,52,46,38,45,68)(16,47,49,53,48,39,18,40)(28,61,64,54,62,67,50,63)(30,55,65,59,43,57,36,66);(2,4)(5,8)(6,15)(7,22)(9,30)(10,31)(11,36)(12,32)(13,37)(14,38)(16,28)(17,45)(18,50)(19,46)(20,51)(21,52)(23,55)(24,56)(25,59)(26,29)(27,60)(33,57)(34,43)(35,65)(39,61)(40,54)(41,44)(42,68)(47,67)(48,62)(49,64)(53,63)(58,66);(5,9,13,35,12,34,10,11)(6,16,20,49,19,48,17,18)(7,23,27,25,26,33,24,58)(8,30,37,65,32,43,31,36)(14,39,42,40,41,47,21,53)(15,28,51,64,46,62,45,50)(22,55,60,59,29,57,56,66)(38,61,68,54,44,67,52,63)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,d^2=ei,e^2=i,f^2=hi,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=fh,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, 413609);