#200891

order	512
nilpotency class	3
isoclinism	isoclinic class of #200737
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

992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=hi,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,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, 108087);