Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12,10,8,7,11)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(1,2,3,4)(5,7,10,9)(6,11,8,12)(14,18)(19,27)(20,25)(21,28);(5,8,10,6)(7,12,9,11)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(6,8)(11,12)(14,18)(19,27)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^c=di,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96669);