Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2050 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(12,20)(15,17)(19,21);(1,3)(2,5)(4,7)(6,8)(9,10,12,16,13,14,20,22)(11,17,19,24,18,15,21,23);(2,6)(5,8)(10,14)(12,20)(19,21)(23,24);(3,7)(5,8)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)Polycyclic presentation:
2^9|b^2=h,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235134, #72235135, #72235136, … |
| maximal normal subgroup of | #72235134, #72235135, #72235136, … |
| maximal quotient of | #72235157, #72235370 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298910);