Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2846 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,20)(19,23,24,22)(26,30)(27,31);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26,29,30)(27,32,31,28);(4,8)(6,7)(10,14)(13,15)(19,24)(22,23)(26,30)(28,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,22,24,23)(25,27)(26,28)(29,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,d^2=ghi,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 | #72336604, #72336612, #72336636 |
| maximal normal subgroup of | #72336604, #72336612, #72336636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2284700);