Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C2×#135 | D4 |
| 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 | |
306 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,6)(3,4)(7,8)(9,10,15,22,17,23,16,36)(11,24,28,40,29,19,35,39)(12,30,27,38,31,37,26,18)(13,32,25,21,14,34,33,20);(1,2)(3,6)(4,8)(5,7)(9,11,15,28,17,29,16,35)(10,18,22,30,23,38,36,37)(12,25,27,14,31,33,26,13)(19,20,39,32,24,21,40,34);(1,3,5,4)(2,6,7,8)(9,12,17,31)(10,19,23,24)(11,25,29,33)(13,28,14,35)(15,27,16,26)(18,20,38,21)(22,39,36,40)(30,32,37,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43553);