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 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(20,27)(24,29)(25,31)(30,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(9,10)(11,15)(12,13)(14,16)(18,22)(23,28)(24,29)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(10,13)(14,15)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgh,d^2=gh,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, 5093473);