Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131979 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,20)(12,24,26,17)(13,18)(14,19)(15,21,31,36)(16,25,32,27)(22,29)(23,34)(28,33)(30,35,40,37)(38,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15,19,31)(12,21,26,36)(13,22,28,38)(16,29,32,39)(17,20,24,34)(18,30,33,40)(25,35,27,37);(3,8)(5,7)(9,12)(10,16)(11,21)(13,27)(14,26)(15,29)(17,18)(19,32)(20,30)(22,37)(23,36)(24,33)(25,28)(31,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284982, #72285119, #72285670, … |
| maximal normal subgroup of | #72284982, #72285119, #72285670, … |
| maximal quotient of | #72339621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39176);