Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1369 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)(9,10,13,14)(11,12,16,15)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(34,38)(35,41)(36,44)(46,48);(9,11)(10,12)(13,16)(14,15)(18,22)(20,28)(23,25)(26,29)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(10,14)(11,16)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,34)(35,39)(36,45)(37,38)(40,44)(41,42)(43,48)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=gh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5063846);