Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10,13,16)(11,17,19,23)(12,15,21,24)(14,20,22,18);(2,3)(7,8)(9,11)(10,14)(12,20,21,18)(13,19)(15,23,24,17)(16,22);(1,3,5,8,4,2,6,7)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,e^2=h,b^a=bd,f^a=fg,c^b=ch,d^b=dg,g^b=gi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35666);