Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,6,4,7,3,8)(9,10,15,21,16,23,32,22)(11,18,26,38,27,35,40,39)(12,28,24,17,30,36,25,37)(13,31,33,19,14,34,29,20);(2,6)(3,5)(7,8)(9,11,14,25,32,40,33,30,16,27,13,24,15,26,29,12)(10,17,19,18,21,36,34,38,23,37,20,35,22,28,31,39);(2,7)(6,8)(9,12,29,26,15,24,13,27,16,30,33,40,32,25,14,11)(10,18,34,37,22,39,19,36,23,35,31,17,21,38,20,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egh,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48473);