Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124679 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1799 |
| prime radical | #392 | Q8 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 436 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)(3,5)(4,7)(6,8)(9,10,13,17,14,18,26,33)(11,19,22,34,23,35,25,39)(12,16,24,30,20,31,37,40)(15,27,28,21,29,38,32,36);(9,11,20,36,14,23,12,21)(10,15,16,19,18,29,31,35)(13,25,37,38,26,22,24,27)(17,32,30,39,33,28,40,34);(2,5)(3,8)(6,7)(13,26)(22,25)(24,37)(27,38)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,f^b=fi,e^c=eh,f^c=fgh,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72264399, #72264443, #72264447, … |
| maximal normal subgroup of | #72264399, #72264443, #72264447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41724);