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 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
678 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 678 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)(9,10,13,18,15,20,14,19)(11,21,24,16,25,31,23,32)(12,26,27,33,29,34,28,17)(22,36,38,35,39,30,37,40);(9,11,14,24,15,25,13,23)(10,16,18,21,20,32,19,31)(12,22,28,38,29,39,27,37)(17,30,26,35,33,36,34,40);(1,3,4,8)(2,5,6,7)(9,12,14,28,15,29,13,27)(10,17,19,34,20,33,18,26)(11,22,24,38,25,39,23,37)(16,30,31,40,32,36,21,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=i,f^2=i,g^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, 15685);