#200758

order	512
nilpotency class	3
isoclinism	isoclinic class of #200746
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	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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:

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

Polycyclic presentation:

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

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, 107954);