Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #59021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1961 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)(13,14)(15,22)(16,24)(17,21)(18,19)(20,25)(23,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,18)(16,23)(19,22)(20,28)(24,27)(25,26);(2,4)(13,15,17,18)(14,19,21,22)(16,23,25,26)(20,27,24,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,24)(22,28);(6,11)(10,12)(14,21)(16,25)(19,22)(23,26)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424436);