Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511751 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1176 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10)(7,13)(8,16)(11,18)(12,19)(14,17)(15,20);(1,2,3,4);(5,6)(7,14)(8,17)(9,11)(10,13)(12,16)(15,19)(18,20);(7,15)(10,18)(13,20)(14,19)Polycyclic presentation:
2^9|b^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,f^c=fh,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226767, C2×#511881 |
| maximal normal subgroup of | #72226767, C2×#511881 |
| maximal quotient of | C2×#511881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419077);