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) = C26 |
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 | |
3005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,23,24,22)(25,26)(27,32)(28,30)(29,31);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(18,21)(19,24)(25,27)(26,28)(29,30)(31,32);(17,19)(18,22)(20,24)(21,23)(25,28)(26,30)(27,31)(29,32);(1,3,5,8)(2,4,6,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,24)(21,22)(25,28)(26,27)(29,32)(30,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=hi,b^2=fgi,d^2=fh,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, 2751347);