Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #34741 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1897 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)(13,14,20,15,19,24,18,23)(16,26,17,22,27,21,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,20,24,19,23,18,14)(16,22,17,21,27,25,28,26);(2,4)(13,16)(14,21)(15,25)(17,20)(18,28)(19,27)(22,23)(24,26);(6,9)(10,12)(13,17)(14,22)(15,21)(16,18)(19,28)(20,27)(23,26)(24,25);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,15)(16,17,27,28)(21,25,26,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6295747);