Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #34988 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1705 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1705 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,14,15,16)(21,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219244, #72255750, #72261983, … |
| maximal normal subgroup of | #72219244, #72261983, #72261987, … |
| maximal quotient of | #72339637, #72339638, #72339645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4300);