Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | #16258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(9,10,13,15)(11,16,12,14)(17,18)(19,22)(20,23)(21,24)(26,28)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,26,29,28)(27,31,32,30);(9,11,13,12)(10,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,6)(5,8)(9,12,13,11)(10,14,15,16)(17,18)(19,21)(20,23)(22,24)(25,28,29,26)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=g,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72303442, #72303510, #72303512, … |
| maximal normal subgroup of | #72303442, #72303510, #72303512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1212760);