Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,12,15,16)(19,25)(20,27)(23,28)(24,29)(33,34)(35,40)(36,43)(37,39)(38,41)(42,47)(44,48)(45,46);(1,3)(2,4)(5,8)(6,7)(10,14)(11,15)(18,22)(23,28)(24,29)(30,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4,5,7)(2,3,6,8)(9,11)(10,12)(13,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,2)(3,4)(5,6)(7,8)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4543525);