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 | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,28)(17,22)(18,23)(19,21)(20,26)(25,35)(27,39)(29,33)(30,38)(31,37)(32,36)(34,41)(40,44)(42,43);(5,7)(6,9,10,11)(8,12)(13,15,17,27,19,26,32,41,18,25,31,40,16,29,30,42)(14,20,22,34,21,35,36,44,23,33,37,43,28,24,38,39);(1,3,4,2)(15,25)(20,33)(24,35)(26,29)(27,40)(34,43)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fgi,c^b=ci,d^b=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35495);