Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17,12,18,11,16)(7,19,22,13,23,31,24,32)(8,14,25,34,21,35,29,33)(9,26,20,15,28,30,27,36);(5,7,12,23)(6,13)(8,9,21,28)(10,22,11,24)(14,15)(16,19)(17,31)(18,32)(20,29,27,25)(26,33)(30,34)(35,36);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,29)(12,21)(13,30)(15,19)(16,33)(17,34)(18,35)(22,28)(23,27)(26,32)(31,36);(5,9,10,20,12,28,11,27)(6,15,16,26,18,36,17,30)(7,21,22,29,23,8,24,25)(13,14,19,33,32,35,31,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272219);