Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C23 | #407 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 94 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)(4,8)(5,6)(9,10,15,11,14,19,32,23,16,20,33,24,13,18,31,22)(12,25,28,37,27,17,34,21,29,35,39,26,30,36,40,38);(1,3,7,2,5,8,4,6)(9,11,15,18,14,23,32,10,16,24,33,19,13,22,31,20)(12,26,39,36,27,38,40,25,29,37,28,17,30,21,34,35);(2,6)(3,8)(9,12,14,27,16,29,13,30)(10,17,18,25,20,36,19,35)(11,21,22,37,24,38,23,26)(15,28,32,34,33,39,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cgh,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239344, #72239374, #72239377 |
| maximal normal subgroup of | #72239344, #72239374, #72239377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49120);