#500727

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,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, 407923);