Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(10,11)(13,14)(15,18)(16,19)(17,21)(20,24)(22,25)(23,27)(26,28);(5,6)(7,8)(9,10,12,11)(13,15,17,23)(14,18,21,27)(16,22,24,28)(19,25,20,26);(2,4)(6,8);(9,11,12,10)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27)Polycyclic presentation:
2^9|b^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264138, #72264231, #72264246, … |
| maximal normal subgroup of | #72264138, #72264231, #72264246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302272);