#148133

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
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:

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

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=fh,h^a=hi,c^b=cf,e^b=eh,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, 55329);