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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,12)(13,16)(14,15)(19,25)(20,27)(23,28)(24,29)(33,34,39,41)(35,37,36,43)(38,40,46,48)(42,45,44,47);(1,3)(2,6)(4,5)(7,8)(10,15)(11,13)(18,22)(23,28)(24,29)(30,32)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,16,15,12)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,4,5,3)(2,6,7,8)(9,13,14,11)(10,16,15,12)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=fghi,d^2=fh,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, 5636241);