#199156

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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

1002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106352);