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 | |
1445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10)(11,14)(12,16)(13,15)(19,25)(20,27)(23,28)(24,29)(33,34,37,38)(35,39,41,45)(36,40,44,43)(42,46,47,48);(9,11,13,12)(10,14,15,16)(18,22)(23,28)(24,29)(30,32)(34,38)(35,41)(40,43)(42,47);(1,2,4,6)(3,5,7,8)(9,12,13,11)(10,14,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgi,d^2=h,e^2=f,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, 4542899);