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 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,12)(13,14)(15,16)(18,22)(19,25)(20,28)(30,32)(36,43)(39,46)(41,47)(44,48);(2,6)(3,8)(10,14)(11,16)(18,22)(20,28)(23,26)(27,31)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,15,14,12)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4,5,7)(2,3,6,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=gi,d^2=ghi,e^2=fh,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, 5791316);