Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#127 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
532 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 174 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:
(2,4)(6,9)(7,10)(11,12)(13,14,16,20,17,21,24,27,18,15,19,22,25,23,26,28);(5,6,10,12,8,11,7,9)(13,15,16,22,17,23,24,28,18,14,19,20,25,21,26,27);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47709);