Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1744 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,26,23,25,21,18)(20,27,29,32,30,31,28,24);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,28,23,30,22,29)(18,24,25,31,26,32,19,27);(9,11,12,15)(10,14,13,16);(17,21,23,22)(18,19,26,25)(20,28,30,29)(24,27,32,31)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343056);