Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2540 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)(4,5)(6,8)(10,13)(11,15)(17,18)(19,23)(20,24)(21,22)(25,26,30,32)(27,29,28,31);(2,5)(3,8)(9,10,12,13)(11,16,15,14)(17,18,21,22)(19,23,24,20)(25,27,30,28)(26,31,32,29);(3,8)(6,7)(11,15)(14,16)(18,22)(20,23)(25,27,30,28)(26,31,32,29);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,22)(21,24)(25,28,30,27)(26,29,32,31);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,24,22,19)(25,29)(26,28)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3814425);