Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,22,24,23)(25,26,29,31)(27,28,32,30);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,30,31,28);(4,8)(6,7)(10,14)(13,15)(18,21)(22,23)(26,31)(27,32);(1,3,5,2)(4,6,8,7)(9,11)(10,15)(12,16)(13,14)(17,19,20,24)(18,22,21,23)(25,28)(26,27)(29,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3408498);