#148361

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
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

661 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)(3,16,21,47)(4,23,28,44)(5,30)(6,13)(7,33)(8,35)(9,15)(10,18,39,53)(11,26,43,29)(12,31)(14,32)(17,48)(19,46,22,38)(20,55,54,37)(24,57)(25,45,34,41)(27,42,59,60)(36,49)(40,51)(50,64)(52,61)(56,63)(58,62);(1,3,5,18,8,22,31,54,9,21,32,53,6,19,7,20)(2,10,12,16,13,37,30,46,15,39,33,47,35,55,14,38)(4,24,25,52,29,56,27,50,28,58,34,17,26,49,59,51)(11,40,41,63,23,48,42,57,43,64,45,36,44,61,60,62);(1,4)(2,11)(3,17)(5,27)(6,29)(7,34)(8,26)(9,28)(10,36)(12,42)(13,44)(14,45)(15,43)(16,40)(18,24)(19,51)(20,56)(21,52)(22,50)(23,35)(25,31)(30,41)(32,59)(33,60)(37,57)(38,48)(39,63)(46,61)(47,64)(49,54)(53,58)(55,62)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,g^b=gi,d^c=dh,e^c=ei,f^c=fh,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, 55557);