#502149

order	512
nilpotency class	4
isoclinism	isoclinic class of #502149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25108
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fh,g^c=ghi,h^c=hi,e^d=ei

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, 409345);