Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3374 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)(3,5,7,8)(9,10,12,13)(11,14,15,16)(18,21)(19,23)(26,29)(30,31);(11,15)(14,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,30)(28,29)(31,32);(10,13)(14,16)(19,23)(22,24)(25,26,28,29)(27,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)(25,27,28,32)(26,30,29,31)Polycyclic presentation:
2^9|a^2=fi,c^2=h,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72336770, #72336786, #72336797, … |
| maximal normal subgroup of | #72336770, #72336786, #72336797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5500660);