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 | |
1485 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)(19,23)(20,22)(21,24)(25,27)(26,30)(28,29)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,5)(6,8)(11,15)(13,16)(18,20)(19,25)(21,29)(30,32)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(9,11)(10,13)(12,15)(14,16)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fg,e^2=g,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, 5865680);