Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| 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 | |
2518 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)(7,17)(8,22,24,13)(9,16)(10,14)(11,15)(12,19)(18,32,34,36)(20,28)(21,27)(23,30,25,31)(26,35,29,33);(2,4)(5,7,10,21,11,20,9,19)(6,12,16,28,15,27,14,17)(8,18,25,35,24,34,23,33)(13,29,31,32,22,26,30,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,22)(16,31)(17,32)(19,33)(20,34)(21,35)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237175, #72237180, #72237186, … |
| maximal normal subgroup of | #72237175, #72237180, #72237186, … |
| maximal quotient of | #72239037, #72239758, #72239764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30733);