Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3028 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,23)(21,22)(25,26,29,31)(27,32,28,30);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19,22,20)(18,23,21,24)(25,27,29,28)(26,30,31,32);(4,8)(6,7)(11,15)(13,16)(18,21)(23,24)(27,28)(30,32);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,24,21,23)(25,26)(27,32)(28,30)(29,31);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,13,14,16)(17,21,22,18)(19,24,20,23)(25,28,29,27)(26,30,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2263480);