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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(13,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(3,8)(5,7)(9,10,12,14)(11,16,15,13)(18,22)(19,25)(20,28)(30,32)(34,35)(36,43)(40,42)(44,48);(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,2)(3,5)(4,6)(7,8)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=f,d^2=gh,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, 6080615);