Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(34,35)(36,43)(37,45)(47,48);(4,8)(6,7)(18,22)(20,28)(23,25)(26,29)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,48,46,47);(2,3)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,48,46,47);(1,3,5,2)(4,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,4,5,8)(2,6,3,7)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,37,38,45)(34,40,35,42)(36,44,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=fghi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5638481);