Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1876 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)(3,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26,30,29)(27,31,32,28);(1,2,5,4)(3,6,7,8)(10,13)(15,16)(18,22)(19,24)(25,27)(26,28)(29,31)(30,32);(3,7)(6,8)(9,11,14,12)(10,15,13,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,31)(26,27,29,32);(1,3,5,7)(2,6,4,8)(9,12,14,11)(10,15,13,16)(17,20,21,23)(18,19,22,24)(25,29,30,26)(27,31,32,28);(1,4,5,2)(3,6,7,8)(9,13,14,10)(11,16,12,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=gh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5588692);