Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124874 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1560 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8585 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 110 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:
(1,2,6,12,23,10,17,26,7,13,16,8,5,11,3,14)(4,18,20,25,19,9,24,27,22,28,15,30,21,29,32,31);(1,3,5,16,7,17,23,6)(2,8,10,14,13,12,11,26)(4,15,19,32,22,20,21,24)(9,25,18,31,29,30,28,27);(1,4)(2,9)(3,15)(5,19)(6,24)(7,22)(8,25)(10,18)(11,28)(12,30)(13,29)(14,31)(16,32)(17,20)(21,23)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=i,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41771);