Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1496 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)(3,7)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,46)(45,48);(2,5)(6,8)(11,15)(13,16)(18,20)(21,29)(23,25)(26,32)(33,35,38,36)(34,39,41,42)(37,44,45,43)(40,47,48,46);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,48,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,41,48)(35,43,36,44)(39,46,42,47)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5865689);