#501657

order	512
nilpotency class	4
isoclinism	isoclinic class of #501655
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	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fh,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, 408853);