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 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(12,16)(14,15)(18,22)(23,25)(24,29)(30,31)(33,34,38,35)(36,39,44,41)(37,40,46,42)(43,47,45,48);(1,3,6,5)(2,7,4,8)(9,10,13,11)(12,14,16,15)(20,28)(24,29)(26,32)(30,31)(34,35)(39,41)(40,42)(47,48);(1,4,6,2)(3,7,5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,34)(36,39,44,41)(37,42,46,40)(43,47,45,48);(1,3,6,5)(2,8,4,7)(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,39)(35,41)(37,45)(38,44)(40,48)(42,47)(43,46);(1,5,6,3)(2,7,4,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,35,42)(36,43,44,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,d^2=fgh,e^2=gh,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, 5681327);