#201859

order	512
nilpotency class	3
isoclinism	isoclinic class of #201717
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1005 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:

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

Polycyclic presentation:

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

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, 109055);