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 | |
1491 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)(6,8)(11,15)(14,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(34,38)(39,41)(40,44)(46,47);(2,5)(6,8)(10,13)(14,16)(18,19)(23,25)(24,26)(30,31)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=ghi,d^2=f,e^2=fg,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, 4541333);