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) = C25×C4 |
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 | |
3099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,16,12,13)(17,18,21,24)(19,23,20,22)(25,26,29,31)(27,32,28,30);(1,3,4,8)(2,5,6,7)(10,15)(11,12)(17,19,21,20)(18,22,24,23)(25,26)(27,30)(28,32)(29,31);(2,6)(5,7)(9,11,14,12)(10,13,15,16)(17,19,21,20)(18,23,24,22)(25,27,29,28)(26,30,31,32);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,13,15,16)(17,20,21,19)(18,23,24,22)(25,28,29,27)(26,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=gi,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, 4621405);