Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
906 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 354 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) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,7,8)(10,13)(11,15)(14,16)(18,21)(19,23)(22,24);(1,2)(3,4)(9,10,11,13,12,14,15,16);(2,4)(17,18,19,21,20,22,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf104 conjugacy classes — view the full character table.
| maximal subgroup of | #72211231, C2×#121387, #72211642, … |
| maximal normal subgroup of | #72211231, C2×#121387, #72211642, … |
| maximal quotient of | #72211228, C2×#121387, #72224857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28583);