Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126723 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1736 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #7895 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,14,24,29,8,15,23,11,6,13,4,21)(3,10,17,27,19,31,16,28,20,32,25,26,18,30,22,9);(1,3)(2,9,13,26,15,28,14,27)(4,16)(5,25)(6,18)(7,19)(8,20)(10,29,30,12,32,21,31,11)(17,23)(22,24);(1,3)(2,10)(4,22)(5,17)(6,18)(7,19)(8,20)(9,21)(11,26)(12,27)(13,30)(14,31)(15,32)(16,24)(23,25)(28,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224951, #72224957, #72224963, … |
| maximal normal subgroup of | #72224951, #72224957, #72224963, … |
| maximal quotient of | C2×#126851 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34047);