Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3718 conjugacy classes of subgroups; only the 570 normal subgroups are shown.
This group has 3718 conjugacy classes of subgroups — too many to draw. Only its 570 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 — 4 generators on 24 points:
(6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24);(2,4)(5,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(11,18)(14,22)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265061);