Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2740 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)(3,7)(4,8)(5,6)(10,11)(12,16)(17,18)(19,23)(20,21)(22,24)(27,28)(30,31);(2,6)(4,7)(10,11)(14,15)(18,20)(19,24)(25,26,29,32)(27,30,28,31);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,27,29,28)(26,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,7)(2,8,6,3)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2868025);