Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,7,10,8,9,11,12)(13,14,18,24,20,23,33,22,19,25,32,41,17,34,16,31)(15,26,29,42,28,21,36,38,30,40,44,37,35,39,27,43);(1,2,3,4)(6,9)(7,11)(13,15)(14,21)(16,27)(17,35)(18,29)(19,30)(20,28)(22,37)(23,40)(24,38)(25,39)(26,34)(31,42)(32,44)(33,36)(41,43);(13,16,17,32,19,33,20,18)(14,22,34,24,25,31,23,41)(15,27,35,44,30,36,28,29)(21,37,26,38,39,42,40,43)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,d^b=di,e^b=egi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34165);