Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #127 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 370 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(11,12)(15,16)(18,21)(19,22)(23,24);(2,4)(9,10,12,15,13,16,11,14);(6,8)(17,18,22,24,20,23,19,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72214336, #72214375, #72214432, … |
| maximal normal subgroup of | #72214336, #72214375, #72214432, … |
| maximal quotient of | #72224886, #72224888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17459);