Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C27 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
11578 conjugacy classes of subgroups; only the 1682 normal subgroups are shown.
This group has 11578 conjugacy classes of subgroups — too many to draw. Only its 1682 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) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,20,21)(19,23,24,22);(1,2,3,4)(5,7,8,6)(10,13)(14,15)(18,21)(22,23);(17,18)(19,22)(20,21)(23,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df152 conjugacy classes — view the full character table.
| maximal subgroup of | #72241120, #72241125, #72241145, … |
| maximal normal subgroup of | #72241120, #72241125, #72241145, … |
| maximal quotient of | #72266813, #72329846, #72331458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420974);