Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(13,14)(15,23)(16,19)(17,22)(18,21)(20,25)(24,28)(26,27);(1,2,3,4)(9,10)(11,12)(14,19)(20,27)(21,22)(23,28);(6,8)(10,12)(13,15,16,24)(14,20,19,27)(17,26,18,25)(21,28,22,23);(14,21)(17,18)(19,22)(20,28)(23,27)(25,26)Polycyclic presentation:
2^9|b^2=ei,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264150, #72264172, #72264302, … |
| maximal normal subgroup of | #72264150, #72264172, #72264302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301443);