Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2396 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)(11,16)(12,13)(14,15)(17,18,20,22)(19,23,24,21)(25,26,30,28)(27,31,29,32);(1,2,4,6)(3,5,7,8)(10,15)(11,12)(17,18,20,22)(19,21,24,23)(25,27,30,29)(26,31,28,32);(3,7)(5,8)(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24)(25,27,30,29)(26,32,28,31);(1,2)(3,8)(4,6)(5,7)(9,12,14,11)(10,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,26)(27,31,29,32);(1,3,4,7)(2,5,6,8)(9,13)(10,12)(11,15)(14,16)(17,19)(18,21)(20,24)(22,23)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3304411);