Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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 | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3556 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)(6,9)(7,11)(10,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(6,10)(9,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(5,6)(7,12)(8,10)(9,11)(15,22)(18,21)(23,28)(25,27);(14,19)(18,21)(20,26)(25,27);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281845);