Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C23 | #478 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(17,18,22,26,23,28,38,27)(19,24,30,40,31,41,21,36)(20,32,33,39,35,47,34,48)(25,42,43,46,44,37,45,29);(1,3,6,13,7,15,5,14)(2,8,9,16,4,12,10,11)(17,19,22,30,23,31,38,21)(18,24,26,40,28,41,27,36)(20,29,33,42,35,46,34,37)(25,39,43,47,44,48,45,32);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(17,20,23,35)(18,25,28,44)(19,29,31,46)(21,37,30,42)(22,33,38,34)(24,39,41,48)(26,43,27,45)(32,40,47,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eg,d^c=dg,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43756);