Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,12,18,14,16,13,17)(11,19,21,23,22,24,20,15);(9,11,13,20,14,22,12,21)(10,15,17,24,16,23,18,19);(2,4);(10,16)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72252110, #72252112, #72252113, … |
| maximal normal subgroup of | #72252110, #72252112, #72252113, … |
| maximal quotient of | #72321990 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260599);