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 | |
1508 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)(9,10,12,15)(11,14,16,13)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,29)(31,32)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(2,5)(6,8)(9,11,12,16)(10,13,15,14)(18,20)(19,25)(21,29)(30,32)(35,40)(36,39)(43,48)(44,47);(9,10)(11,14)(12,15)(13,16)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=gh,d^2=fg,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, 5870653);