Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129558 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C23 | C2×#104 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,17,13,20,22,16,14,18,23,24,21,19,11,15);(2,5)(6,7)(10,15,19,24,18,16,20,17);(1,3,4,8)(2,6,5,7)(9,11,13,12,14,22,21,23)(10,16,20,24,18,15,19,17)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cghi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279590, #72280278, #72280329 |
| maximal normal subgroup of | #72279590, #72280278, #72280329 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49125);