#199140

order	512
nilpotency class	3
isoclinism	isoclinic class of #198834
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)(3,13,17,24,16,23,15,8)(4,18,19,26,21,9,20,25)(14,27,28,22,30,32,29,31)(33,34)(35,40,47,55)(36,48)(37,44)(38,42)(39,43)(41,49)(45,59,61,63)(46,57,51,54)(50,58)(52,56)(53,60,64,62);(1,3)(2,8,12,24)(4,14)(5,15)(6,17)(7,16)(9,22,18,31)(10,23,11,13)(19,28)(20,29)(21,30)(25,32,26,27)(33,35)(34,40,43,55)(36,45)(37,51)(38,46)(39,47)(41,53,56,64)(42,57,44,54)(48,59,58,63)(49,62)(50,61)(52,60);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,26)(12,18)(13,27)(15,28)(16,30)(17,29)(23,32)(24,31)(33,36,39,50)(34,41,43,56)(35,45,47,61)(37,49,38,52)(40,53,55,64)(42,48,44,58)(46,60,51,62)(54,63,57,59);(1,5,7,6)(2,10,12,11)(3,15,16,17)(4,19,21,20)(8,23,24,13)(9,25,18,26)(14,28,30,29)(22,32,31,27)(33,37,39,38)(34,42,43,44)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,61,62)(53,63,64,59)

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=di,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, 106336);