Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,7,8)(13,14,17,18)(15,19,23,21)(16,24,25,20)(22,27,28,26);(9,10,11,12)(14,18)(19,21)(20,24)(26,27);(5,6)(7,8)(9,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27);(1,3,4,2)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=i,b^2=fh,d^2=f,e^2=gi,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424371);