#200749

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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:

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

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,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, 107945);