Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,16,9,13)(7,17,20,33,21,34,19,31)(8,22,25,32,26,30,24,14)(12,23,28,36,29,35,27,18);(1,2,3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,27)(14,31)(15,28)(16,29)(17,22)(18,24)(25,36)(26,35)(30,34)(32,33);(6,13,16,15)(12,27,29,28)(14,30,32,22)(17,31,34,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250626, #72250659, #72250663, … |
| maximal normal subgroup of | #72250626, #72250659, #72250663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54657);