#148141

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

671 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,g^2=i,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, 55337);