Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4234 conjugacy classes of subgroups; only the 362 normal subgroups are shown.
This group has 4234 conjugacy classes of subgroups — too many to draw. Only its 362 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,26)(27,28);(6,9)(10,12)(15,22)(16,25)(18,21)(20,24);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,23,17,22,25,28)(14,18,20,26,19,21,24,27);(7,11)(10,12)(14,19)(18,21)(20,24)(26,27)Polycyclic presentation:
2^9|c^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318363, #72318621, #72318704 |
| maximal normal subgroup of | #72318363, #72318621, #72318704 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289594);