#148535

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
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

639 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,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, 55731);