Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119716 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,24,21)(22,23);(3,8)(5,6)(9,11)(10,14)(12,21)(13,20)(15,23)(16,18)(17,24)(19,22);(2,5)(3,8)(6,7)(9,11)(10,15,16,24)(12,19)(13,20)(14,17,18,23)(21,22)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232815, #72233074, #72233076, … |
| maximal normal subgroup of | #72232815, #72233074, #72233076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26912);