Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3450 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)(6,7)(10,11)(13,14,19,18)(15,16,17,20)(21,22)(23,26)(24,28)(25,27);(5,6,9,7)(8,11,12,10)(13,15,19,17)(14,20,18,16)(21,23,25,28)(22,26,27,24);(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,17,19,15)(14,20,18,16)(21,24)(22,23)(25,26)(27,28);(5,8,9,12)(6,10,7,11)(13,18,19,14)(15,20,17,16)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 555271);