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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2530 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,23)(20,22)(21,24)(25,26,29,31)(27,32,28,30);(3,7)(5,8)(10,13)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,29,28)(26,30,31,32);(1,2,4,6)(3,5,7,8)(9,11)(10,14)(12,16)(13,15)(17,18,20,22)(19,21,24,23)(25,26,29,31)(27,32,28,30);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23)(25,28,29,27)(26,30,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,e^2=fg,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.
| maximal subgroup of | #72303510, #72303515, #72303520, … |
| maximal normal subgroup of | #72303510, #72303515, #72303520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5546360);