Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10)(11,25,27,39)(12,30,32,37)(13,21)(14,20)(15,23)(16,22)(17,24)(18,26,36,33)(19,29,35,28)(31,40)(34,38);(1,2,6,8,4,7,3,5)(9,11,14,32,16,29,13,33,17,27,31,12,15,28,34,26)(10,18,21,19,23,37,20,39,24,36,38,35,22,30,40,25);(9,12,31,28,16,26,34,11,17,32,14,29,15,33,13,27)(10,19,38,37,23,39,40,36,24,35,21,30,22,25,20,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224819, #72224823, #72224824, … |
| maximal normal subgroup of | #72224819, #72224823, #72224824, … |
| maximal quotient of | #72225253, #72225254, #72225287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48955);