Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151066 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#367 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #367 |
| LCS | G ▷ C2×C16 ▷ 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 | |
132 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 132 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,23,25,36)(19,28)(20,32)(21,33)(24,39)(26,40,43,48)(27,46,44,38)(29,34)(30,42)(31,41)(35,47,45,37);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,24,40,20,26,19,23,33,44,28,37,29,35,39,46,22,25,42,48,34,43,31,36,21,27,41,47,32,45,30,38);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=dfh,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,e^b=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232028, #72234329, #72234330, … |
| maximal normal subgroup of | #72232028, #72234329, #72234330, … |
| maximal quotient of | #72286713, #72322668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58262);