#499340

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,g^b=gi,d^c=di,e^c=eh,f^c=fhi,g^c=gh,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, 406536);