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 | |
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)(3,7)(11,15)(14,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(2,5)(6,8)(10,13)(14,16)(33,35,38,36)(34,39,41,42)(37,44,46,43)(40,48,45,47);(9,10,12,13)(11,14,15,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,36,38,35)(34,39,41,42)(37,43,46,44)(40,48,45,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=g,e^2=g,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, 5868522);