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 | |
1398 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)(10,12)(11,15)(17,18)(19,23)(20,22)(21,28)(24,29)(25,31)(26,27)(30,32)(34,38)(36,44)(39,41)(42,48);(3,8)(5,7)(18,20)(21,29)(23,27)(25,32)(33,34,37,38)(35,41,43,39)(36,45,44,40)(42,46,48,47);(9,10,13,12)(11,16,15,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,12,16)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,32,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,32)(27,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=gh,d^2=fgh,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, 5908304);