Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #2403 | C4 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1360 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,15,21,23)(11,17,19,24,18,14,20,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,20)(13,18)(15,17)(16,24)(19,21)(22,23);(1,2)(3,5)(4,6)(7,8)(14,17)(22,24)Polycyclic presentation:
2^9|a^2=g,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232790, #72234080, #72234091, … |
| maximal normal subgroup of | #72232790, #72234080, #72234091, … |
| maximal quotient of | #72232785, #72233226, #72257805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18657);