Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(2,4)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(5,6,7,8)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224736, #72258151, #72280505, … |
| maximal normal subgroup of | #72224736, #72258151, #72280505, … |
| maximal quotient of | #72224729, #72258168, #72258174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261249);