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 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,11,14,12)(10,15,16,13)(17,19,21,24)(18,22,20,23)(26,31)(27,29);(3,8)(6,7)(10,16)(11,12)(18,20)(22,23)(25,27,30,29)(26,32,31,28);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,15,16,13)(17,20,21,18)(19,23,24,22)(25,28)(26,27)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,13,14,15)(10,11,16,12)(17,19)(18,22)(20,23)(21,24)(25,29,30,27)(26,32,31,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=gi,d^2=i,e^2=h,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5846276);