Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,11,12,9)(8,10)(13,14)(15,23,26,28)(16,19)(17,21)(18,22)(20,25,27,24);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,22,27)(16,25,17,24)(19,23,21,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13260);