Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,16,12,14)(17,18)(19,22)(20,23)(21,24)(26,28)(30,31);(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,24)(22,23)(26,28)(30,31);(2,3)(6,7)(10,15)(11,12)(19,24)(21,22)(25,26,29,28)(27,31,32,30);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,16,15,14)(17,18)(19,21)(20,23)(22,24)(25,27,29,32)(26,30,28,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fh,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4086926);