#148140

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi

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, 55336);