Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5426 conjugacy classes of subgroups; only the 1898 normal subgroups are shown.
This group has 5426 conjugacy classes of subgroups — too many to draw. Only its 1898 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) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(9,11)(10,13)(12,16)(14,15);(5,6,7,8)(11,16)(13,15)(18,21)(22,24);(10,14)(13,15);(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf176 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267389 |
| maximal subgroup of | #72258568, #72267389, #72267413, … |
| maximal normal subgroup of | #72258568, #72267389, #72267413, … |
| maximal quotient of | #72258449, #72331622, C2×#513451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420647);