Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,11)(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);(6,9,11,10)(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)(14,20)(15,24)(21,27)(22,33)(25,37)(26,38)(34,40)(39,44)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgi,c^b=ci,d^b=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35720);