Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,4,3)(5,6)(7,8)(9,11)(10,12)(13,14)(15,23)(16,27)(17,22)(18,20)(19,24)(21,29)(25,31)(26,36)(28,40)(30,33)(32,38)(34,41)(35,37)(39,42)(43,44);(5,7,9,12,10,11,8,6)(13,15,16,25,17,26,28,39,18,24,29,37,30,38,41,44)(14,19,21,31,22,32,34,42,20,23,27,35,33,36,40,43);(1,3,4,2)(5,8,10,9)(6,7,12,11)(14,20)(19,23)(21,27)(22,33)(31,35)(32,36)(34,40)(42,43)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgi,c^b=cg,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35760);