#200365

order	512
nilpotency class	3
isoclinism	isoclinic class of #200146
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

996 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh

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, 107561);