Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511887 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 960 conjugacy classes of subgroups — too many to draw. Only its 190 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)(6,9)(7,12)(8,10)(13,14)(15,20)(16,19)(17,18);(5,6,10,13,11,14,8,9)(7,12,18,20,16,19,15,17);(1,2,3,4);(5,7)(6,12)(8,15)(9,17)(10,18)(11,16)(13,20)(14,19)Polycyclic presentation:
2^9|b^2=ei,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=gh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512372, #72226767 |
| maximal normal subgroup of | C2×#512372, #72226767 |
| maximal quotient of | C2×#512372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419568);