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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(12,16)(14,15)(18,22)(23,25)(24,29)(30,31)(33,34,37,38)(35,39,42,45)(36,40,43,46)(41,47,44,48);(3,7)(5,8)(9,10,13,11)(12,14,16,15)(20,28)(24,29)(26,32)(30,31)(34,38)(36,43)(39,45)(41,44);(1,3,4,7)(2,5,6,8)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,42)(34,39,38,45)(36,44,43,41)(40,48,46,47);(1,2)(3,5)(4,6)(7,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,37,43)(34,40,38,46)(35,41,42,44)(39,47,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=fghi,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, 5809005);