Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1402 |
| commutator | C4⋊Q8 | C2×C8 |
| Frattini, cosocle | #1962 | C22 |
| socle | C2 | #7375 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1962 ◁ G | |
| LECS | G ▷ #1962 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 32 points:
(1,2,5,10,6,12,22,29,7,13,21,9,24,11,4,19)(3,14,15,26,16,30,20,27,18,31,23,25,17,28,32,8);(1,3)(2,8,11,25,13,27,12,26)(4,20)(5,23)(6,16)(7,18)(9,28,19,14,10,30,29,31)(15,21)(17,24)(22,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cg,d^b=deh,g^b=gh,d^c=dgh,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1794);