Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,18)(12,26)(13,29)(14,31)(15,33)(16,21)(17,22)(19,27)(20,30)(23,35)(24,40)(25,37)(28,34)(32,38)(36,39);(2,4)(9,11,14,23,30,39,27,12,16,25,13,22,15,24,32,28)(10,17,19,18,33,26,31,40,21,35,38,37,20,34,29,36);(9,12,27,24,15,23,14,25,16,28,32,39,30,22,13,11)(10,18,29,17,20,36,38,34,21,37,31,35,33,40,19,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286901, #72286911, #72287485, … |
| maximal normal subgroup of | #72286901, #72286911, #72287485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54290);