Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,11,12,10,16,9,15)(7,13,17,24,19,26,20,28)(8,14,21,25,23,29,22,30)(18,27,31,34,32,35,33,36);(6,12)(7,17)(8,21)(9,11)(13,26)(14,29)(15,16)(19,20)(22,23)(27,34)(31,33)(35,36);(1,2,3,4)(5,7)(6,13)(8,18)(9,20)(10,19)(11,17)(12,24)(14,27)(15,28)(16,26)(21,31)(22,33)(23,32)(25,34)(29,35)(30,36);(5,8)(6,14)(7,18)(9,22)(10,23)(11,21)(12,25)(13,27)(15,30)(16,29)(17,31)(19,32)(20,33)(24,34)(26,35)(28,36);(7,19)(13,26)(17,20)(18,32)(24,28)(27,35)(31,33)(34,36)Polycyclic presentation:
2^9|a^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350231);