Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12433 |
| 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 | |
305 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 305 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)(9,10,14,19,15,21,31,20)(11,17,24,34,25,36,40,35)(12,26,28,33,29,32,23,16)(13,30,27,18,22,37,39,38);(2,6)(3,5)(7,8)(9,11,22,28,31,40,27,12,15,25,13,23,14,24,39,29)(10,16,18,17,19,26,37,34,21,33,38,36,20,32,30,35);(2,7)(6,8)(9,12,27,24,14,28,22,25,15,29,39,40,31,23,13,11)(10,17,30,16,20,35,38,32,21,36,37,33,19,34,18,26)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,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48474);