Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #44887 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(14,19)(15,22)(21,27)(25,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,19)(15,23,22,28)(16,21,18,27)(20,25,24,26);(2,4)(13,14)(15,24)(16,27)(17,19)(18,21)(20,22)(23,25)(26,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,22)(14,20,19,24)(16,26,18,25)(21,28,27,23);(6,11)(10,12)(13,16,17,18)(14,21,19,27)(15,25,22,26)(20,23,24,28)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424753);