Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,9,11,8,10,12,7)(13,14,17,23,19,25,15,21,20,26,16,22,18,24,27,28);(1,2,3,4)(6,10)(7,11)(14,21)(15,16)(17,27)(18,19)(22,24)(23,25)(26,28);(5,7,12,10,8,11,9,6)(13,15,18,17,20,27,19,16)(14,21,24,23,26,28,25,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36348);