Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(2,5)(3,7)(6,8)(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(1,2,3,5,4,6,7,8);(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^c=ehi,f^c=fi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216361, #72225155, #72225231, … |
| maximal normal subgroup of | #72216361, #72225155, #72225231, … |
| maximal quotient of | #72216287, #72227256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42136);