Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 72 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)(17,18,19,25,29,40,44,36,21,27,30,41,33,38,45,34,22,24,31,37,46,48,47,42,32,39,43,35,20,26,28,23);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(19,28)(20,29)(21,32)(24,34)(25,26)(27,42)(30,47)(31,45)(33,46)(35,40)(36,39)(37,38)(41,48)(43,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,24)(23,34)(25,37)(26,38)(27,39)(35,41)(36,42)(40,48)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,c^a=ci,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58165);