Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13429 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11087 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,12,16,15)(4,17)(9,26,28,32)(10,23)(11,25)(13,31,29,22)(14,27,18,30)(19,21)(20,24);(1,2,4,8,7,11,17,25)(3,13,14,26,16,29,18,32)(5,10,19,23,6,20,21,24)(9,27,22,12,28,30,31,15);(1,3)(2,9)(4,18)(5,15)(6,12)(7,16)(8,22)(10,29)(11,28)(13,20)(14,17)(19,30)(21,27)(23,32)(24,26)(25,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55463);