Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,12,15)(11,13,16,14)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,38,36)(35,39,41,43)(37,44,46,40)(42,48,45,47);(2,5)(3,7)(9,11,12,16)(10,13,15,14)(20,28)(23,30)(26,32)(29,31)(34,36)(35,41)(40,44)(42,45);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,14,15,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,39)(36,43)(37,45)(38,41)(40,48)(42,46)(44,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,34)(35,39,41,43)(37,44,46,40)(42,47,45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=i,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176403);