#200147

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

970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=i,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, 107343);