Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C4⋊C4 | C22 |
| socle | C23 | #342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C4⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 24 points:
(1,2,4,7,5,8,3,6)(18,21)(19,23)(22,24);(2,6,8,7)(9,10,11,13,12,14,15,16)(17,18)(19,21,23,24)(20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=fi,e^2=h,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 595);