Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,13,12)(11,14,16,15)(18,22)(19,25)(20,28)(30,32)(33,34,38,39)(35,36,42,40)(37,41,46,47)(43,44,45,48);(3,7)(6,8)(9,11,13,16)(10,14,12,15)(18,22)(20,28)(23,26)(27,31)(34,39)(35,42)(41,47)(43,45);(2,5)(6,8)(10,12)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=ghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5429536);