Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(9,10,11,12)(13,14)(15,22)(16,17)(18,19)(20,25)(21,27)(23,24)(26,28);(6,8)(9,10,11,12)(14,19)(17,21)(20,22)(24,28);(5,6,7,8)(9,10)(11,12)(13,15,18,25)(14,20,19,22)(16,26,27,23)(17,24,21,28);(1,2,3,4)(13,16)(14,21)(15,23)(17,19)(18,27)(20,24)(22,28)(25,26);(13,17,18,21)(14,16,19,27)(15,24,25,28)(20,26,22,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=gh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 626931);