Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2615 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,17,18,20,16,22)(15,21,24,25,23,27,26,28);(5,6)(7,10)(8,9)(11,12)(14,20)(15,23)(17,22)(24,26);(2,4)(6,9)(10,12)(13,15)(14,21)(16,26)(17,25)(18,23)(19,24)(20,27)(22,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,22,20,17)(15,24,23,26)(21,25,27,28);(13,17,19,20,18,22,16,14)(15,25,24,27,23,28,26,21)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=cf,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285412);