Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1500 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)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,25,29)(20,24,27,28)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(3,7)(5,8)(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,32)(30,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,29)(20,26,27,31)(24,30,28,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,27)(22,28)(23,30)(25,31)(29,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fhi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5156810);