#502136

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
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	#24280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

731 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)(5,6,10,16,32,46,34,48,12,18,31,45,9,15,11,17)(7,13,21,37,54,65,22,38,23,39,56,67,20,36,33,47)(8,24,27,41,26,59,61,44,30,60,62,43,29,42,28,14)(19,49,51,63,50,40,25,58,53,68,57,64,52,66,55,35);(2,4)(5,7,10,21,32,54,34,22,12,23,31,56,9,20,11,33)(6,13,16,37,46,65,48,38,18,39,45,67,15,36,17,47)(8,25,27,53,26,57,61,52,30,55,62,19,29,51,28,50)(14,40,24,58,41,68,59,64,44,66,60,35,43,49,42,63);(5,8,12,30)(6,14,18,44)(7,19,23,53)(9,26,32,29)(10,28,31,61)(11,27,34,62)(13,35,39,58)(15,41,46,43)(16,42,45,59)(17,24,48,60)(20,50,54,52)(21,55,56,25)(22,57,33,51)(36,63,65,64)(37,66,67,40)(38,68,47,49);(2,4)(5,9,12,32)(6,15,18,46)(7,20,23,54)(8,26,30,29)(10,11,31,34)(13,36,39,65)(14,41,44,43)(16,17,45,48)(19,50,53,52)(21,33,56,22)(24,59,60,42)(25,57,55,51)(27,61,62,28)(35,63,58,64)(37,47,67,38)(40,68,66,49)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,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, 409332);