Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(10,13)(11,15)(17,18,21,25)(19,23,27,22)(20,24,29,32)(26,31,28,30);(6,8)(11,15)(14,16)(17,19,21,27)(18,22,25,23)(20,28,29,26)(24,31,32,30);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,31)(29,32);(1,2,3,4)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,30)(26,29)(31,32);(9,10)(11,14)(12,13)(15,16)(17,20,21,29)(18,24,25,32)(19,26,27,28)(22,30,23,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3042292);