Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490693 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,21)(19,24)(20,23)(25,26)(27,28)(29,32)(30,31);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,18,23,26,22,25,20,21)(19,27,31,30,28,24,29,32);(3,11)(7,15)(10,16)(12,13)(17,19,22,28)(18,24,25,27)(20,29,23,31)(21,30,26,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(19,28)(24,27)(29,31)(30,32)Polycyclic presentation:
2^9|b^2=ei,c^2=i,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^c=di,f^c=fh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285431, #72285451, #72285618, … |
| maximal normal subgroup of | #72285431, #72285451, #72285618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397959);