Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,19,24,29,37,44,35,21,26,30,38,33,41,45,36,22,27,31,39,46,48,47,42,32,40,43,34,20,25,28,23);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(19,28)(20,29)(21,32)(24,25)(26,42)(27,36)(30,47)(31,45)(33,46)(34,37)(35,40)(38,48)(39,41)(43,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58161);