Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,4,3,5,6,8,7)(10,15)(12,20)(16,22)(17,24)(18,23)(19,21);(9,10,13,17,14,18,11,16)(12,15,19,22,20,23,21,24);(1,3,4,6,5,7,8,2)(9,11,14,13)(10,16,18,17)(12,19,20,21)(15,22,23,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fgi,d^b=dg,g^b=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35694);