#200733

order	512
nilpotency class	3
isoclinism	isoclinic class of #200728
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

967 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)(3,12)(4,16,19,26)(5,20)(6,10)(7,11)(8,14)(9,18,25,17)(13,28,30,22)(15,23)(21,24)(27,29,32,31)(33,34)(35,44,47,55)(36,48,50,41)(37,51)(38,42)(39,43)(40,46,56,52)(45,59)(49,57,53,58)(54,61)(60,63)(62,64);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,18,31,19,30,17,29)(9,22,26,27,25,28,16,32)(33,35,38,52,39,47,37,46)(34,40,42,55,43,56,51,44)(36,45,53,61,50,62,49,60)(41,54,57,64,48,63,58,59);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,31)(15,30)(16,20)(21,29)(23,32)(24,28)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,48)(44,59)(46,60)(47,62)(51,58)(52,61)(55,64)(56,63);(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,49,50,53)(40,55,56,44)(41,57,48,58)(45,60,62,61)(54,64,63,59)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,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, 107929);