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 | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#536 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 756 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,21)(12,24,27,36)(13,28,30,18)(14,19)(15,20)(16,23,32,29)(17,22,35,25)(26,33)(31,39)(34,37)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11,15,14)(10,16,20,32)(12,25,27,22)(13,23,30,29)(17,33,35,31)(18,21,28,19)(24,34,36,40)(26,38,39,37);(3,8)(5,7)(9,12,15,27)(10,17,20,35)(11,22,14,25)(13,26,30,39)(16,31,32,33)(18,34,28,40)(19,36,21,24)(23,37,29,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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 | #72228029 |
| maximal normal subgroup of | #72228029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39180);