Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| 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 | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,14,16)(11,12,13,15)(17,18)(19,21)(20,22)(23,24)(26,29)(27,28);(2,6)(3,8)(9,11,14,13)(10,15,16,12)(18,19)(20,24)(25,26)(27,31)(28,32)(29,30);(3,8)(4,7)(11,13)(12,15)(17,18,21,19)(20,23,24,22)(25,27,30,28)(26,31,29,32);(1,3,5,8)(2,7,6,4)(9,12,14,15)(10,11,16,13)(17,19,21,18)(20,23,24,22)(25,28,30,27)(26,31,29,32);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,15,16,12)(17,20,21,24)(18,22,19,23)(25,29,30,26)(27,31,28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135190);