Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
6540 conjugacy classes of subgroups; only the 578 normal subgroups are shown.
This group has 6540 conjugacy classes of subgroups — too many to draw. Only its 578 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,8)(9,10,11,12)(14,16)(17,18)(19,22)(20,21)(23,24);(2,4)(6,8)(10,12)(13,14,15,16)(18,21)(22,23);(9,10,11,12)(13,14,15,16)(17,19,20,24)(18,22,21,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72279199 |
| maximal subgroup of | #72264978, #72264988, #72278119, … |
| maximal normal subgroup of | #72264978, #72264988, #72278119, … |
| maximal quotient of | #72265828, #72265892, C2×#158869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 66065);