Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2470 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)(7,12,17,28)(8,20,22,30)(9,15)(10,25)(11,14)(13,23,29,24)(16,31)(18,27,19,34)(21,35)(26,33)(32,36);(2,4)(5,7,10,19,11,17,9,18)(6,12,15,27,14,28,25,34)(8,21,24,32,22,16,23,33)(13,26,20,31,29,36,30,35);(5,8)(6,13)(7,16)(9,24)(10,23)(11,22)(12,26)(14,29)(15,30)(17,21)(18,33)(19,32)(20,25)(27,35)(28,36)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239049, #72252094, #72252103, … |
| maximal normal subgroup of | #72239049, #72252094, #72252103, … |
| maximal quotient of | #72239061, #72239086, #72239110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30832);