Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #354 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #2150 | C22 |
| socle | C2 | #7057 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,6,12,23,34,5,9,24,31,25,32,52,56,7,13,21,35,27,39,49,59,26,11,22,37,50,38,51,60)(3,14,16,40,18,41,45,61,19,30,44,54,47,55,63,64,15,36,42,33,43,57,62,58,17,8,20,28,46,29,48,53);(1,3)(2,8,13,30)(4,20)(5,19)(6,18)(7,15)(9,14,11,36)(10,33,35,40)(12,29,39,55)(16,24)(17,26)(21,44)(22,42)(23,48)(25,47)(27,43)(28,37,54,31)(32,41,38,57)(34,58,59,61)(45,52)(46,50)(49,63)(51,62)(53,60,64,56)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=g,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgi,d^b=degi,e^b=ei,d^c=dgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 896);