#499315

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

742 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,d^b=di,e^c=eh,f^c=fh,g^c=ghi,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, 406511);