Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| 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 | #7062 |
| 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 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,6,13,23,38,8,10,22,34,28,35,51,58,7,14,24,39,27,41,52,61,5,12,25,37,26,40,53,60)(3,15,17,42,19,43,46,62,20,9,29,30,48,31,54,55,16,32,44,36,45,56,63,59,18,33,21,49,47,57,50,64);(1,3,5,18,7,16,8,20)(2,9,12,15,14,33,10,32)(4,21,22,17,24,29,25,44)(6,19,26,47,27,45,28,48)(11,36,34,49,39,42,37,30)(13,31,40,43,41,57,35,56)(23,50,51,46,52,54,53,63)(38,59,58,64,61,62,60,55)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=gi,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=ci,d^b=deg,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 918);