#148518

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,h^b=hi,e^c=ei,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, 55714);