Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2574 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(18,22)(19,24)(25,26)(27,32)(28,31)(29,30);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,12,15,16)(17,18,21,22)(19,23,24,20)(25,27,30,28)(26,31,29,32);(2,4)(6,7)(10,15)(11,13)(18,22)(20,23)(25,28,30,27)(26,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,12,15,16)(17,20,21,23)(18,19,22,24)(25,29,30,26)(27,32,28,31)Polycyclic presentation:
2^9|b^2=f,c^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4538482);