Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106017 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
708 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,20,24,35)(12,25)(13,18)(14,28)(15,19)(16,22,31,23)(17,26)(21,30,37,40)(27,32)(29,33)(34,38,39,36);(9,11)(10,16,19,31)(12,21)(13,22)(14,23)(15,24)(17,30,33,40)(18,20,28,35)(25,39,32,34)(26,36)(27,37)(29,38);(1,3,4,8)(2,5,6,7)(9,12,14,29,15,27,13,26)(10,17,28,25,19,33,18,32)(11,21,23,38,24,37,22,36)(16,30,35,39,31,40,20,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,e^c=ei,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15677);