#200897

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

997 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^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,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, 108093);