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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,16)(14,15)(18,22)(19,20)(25,26,29,28)(27,31,32,30);(3,4)(6,7)(10,15)(11,12)(17,18)(19,24)(20,23)(21,22);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,13,15,16)(17,19,21,20)(18,23,22,24)(25,26,29,28)(27,30,32,31);(1,4,5,3)(2,7,8,6)(9,12,14,11)(10,13,15,16)(17,20,21,19)(18,23,22,24)(25,27,29,32)(26,30,28,31);(1,2)(3,6)(4,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,24)(20,23)(21,22)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,e^2=hi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2503834);