Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,34,38,37)(35,41,42,39)(36,44,46,40)(43,47,45,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(20,28)(30,32)(35,42)(39,41)(43,45)(47,48);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,38,42)(34,39,37,41)(36,45,46,43)(40,48,44,47);(1,2)(3,6)(4,5)(7,8)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,46)(34,40,37,44)(35,43,42,45)(39,47,41,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,34)(35,41,42,39)(36,44,46,40)(43,48,45,47)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5867547);