Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24)(26,30)(28,31);(2,5)(6,8)(11,15)(14,16)(18,21)(22,24)(25,26,29,30)(27,28,32,31);(3,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,22,21,24)(25,27,29,32)(26,31,30,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232590, #72232606, #72232648, … |
| maximal normal subgroup of | #72232590, #72232606, #72232648, … |
| maximal quotient of | #72328521, #72328733, #72330626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2138573);