Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,11,9,7,8,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(5,7,10,12,9,6,8,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,27)(21,22)(26,28);(1,2,3,4)(5,8,9,10)(6,11,7,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,27,22,28);(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,28,22,27);(13,17,18,16)(14,20,21,26)(15,24,25,23)(19,27,22,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6297111);