Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,20)(9,24)(10,15)(11,17)(12,22)(13,28)(14,21)(18,26)(19,35)(23,29)(25,30)(27,36)(31,34)(32,33);(5,7,8,17,21,33,28,36,10,18,23,30,9,19,22,34)(6,11,13,26,24,31,20,32,15,25,12,16,14,27,29,35);(1,3,4,2)(5,6)(7,11)(8,13)(9,14)(10,15)(12,23)(16,30)(17,26)(18,25)(19,27)(20,28)(21,24)(22,29)(31,33)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250186, #72250206, #72250219, … |
| maximal normal subgroup of | #72250186, #72250206, #72250219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54377);