Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2670 conjugacy classes of subgroups; only the 278 normal subgroups are shown.
This group has 2670 conjugacy classes of subgroups — too many to draw. Only its 278 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22);(2,5)(6,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,24,19,22)(16,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15,16,19)(11,17)(13,21)(14,22,23,24)(18,20);(3,7)(6,8)Polycyclic presentation:
2^9|b^2=f,c^2=h,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225654, #72225658, #72225713, … |
| maximal normal subgroup of | #72225654, #72225658, #72225713, … |
| maximal quotient of | #72302092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289903);