Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
18538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,18)(12,17)(13,21)(14,16)(15,19)(20,24)(22,23);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,4);(10,16)(15,23)(17,21)(18,24);(9,12,14,13)(10,17,16,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231659, #72231663, #72231671, … |
| maximal normal subgroup of | #72231659, #72231663, #72231671, … |
| maximal quotient of | #72235637, #72237157, #72238181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277911);