Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,21,24,30)(9,25)(10,14)(11,15)(12,18)(13,23,31,26)(17,27,22,29)(19,28)(20,33)(32,34,36,35);(2,4)(5,7)(6,12,15,28)(8,22,24,17)(9,18)(10,19)(11,20)(13,29)(14,16,25,33)(21,32)(23,35,26,34)(27,31)(30,36);(5,8)(6,13)(7,17)(9,23)(10,26)(11,24)(12,27)(14,30)(15,31)(16,32)(18,34)(19,35)(20,22)(21,25)(28,29)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30808);