#200901

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

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,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, 108097);