Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3032 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)(3,6)(4,7)(5,8)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,23,24,21)(27,28)(30,31);(1,3,4,8)(2,5,7,6)(10,13)(11,16)(17,19,20,24)(18,21,22,23)(25,26,29,32)(27,30,28,31);(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(25,27,29,28)(26,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=hi,b^2=fgi,c^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2974491);