Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
818 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,10,8,9,11,12)(13,14)(15,21,22,18)(16,20)(17,19)(23,27,28,25)(24,26);(13,15,16,23,17,22,24,28)(14,18,20,25,19,21,26,27);(1,2,3,4)(6,9)(10,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=dh,e^b=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12494);