#201957

order	512
nilpotency class	3
isoclinism	isoclinic class of #201948
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

983 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,19)(14,28)(16,24)(17,23)(20,25)(21,26)(22,30)(27,29)(31,32)(33,34)(35,39)(36,46,48,40)(37,49)(38,42)(41,50)(43,51,56,59)(44,57)(45,53)(47,54,61,60)(52,58)(55,63,64,62);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,19,29,20,30,21,31)(9,22,18,27,26,28,25,32)(33,35,38,45)(34,39,42,53)(36,43,48,56)(37,44,50,58)(40,51,46,59)(41,52,49,57)(47,55,61,64)(54,62,60,63);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)(33,36)(34,40)(35,43)(37,47)(38,48)(39,51)(41,54)(42,46)(44,55)(45,56)(49,60)(50,61)(52,62)(53,59)(57,63)(58,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,25,26,18)(14,29,30,31)(22,32,28,27)(33,37,38,50)(34,41,42,49)(35,44,45,58)(36,47,48,61)(39,52,53,57)(40,54,46,60)(43,55,56,64)(51,62,59,63)

Polycyclic presentation:

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

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, 109153);