Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454045 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2098 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(10,13)(14,16)(17,18)(19,23,24,21)(20,22);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16);(3,7)(5,8)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24);(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227413 |
| maximal normal subgroup of | #72227413 |
| maximal quotient of | #72267901, #72268221, #72268314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374306);