Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,4)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,22,24,23)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,2)(3,8)(4,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,24)(18,22,20,23)(27,34)(28,37)(30,33)(31,36);(1,3,6,4)(2,5,7,8)(9,11)(10,13)(12,15)(14,16)(17,20,21,18)(19,22,24,23)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,4,6,3)(2,5,7,8)(17,19)(18,23)(20,22)(21,24)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,5)(2,3)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fgi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6144010);