Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9072 |
| prime radical | #3054 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,13,22,29,8,10,23,11,6,25,4,20)(3,14,17,30,19,31,21,9,16,27,24,28,18,32,15,26);(1,3,6,18,8,16,7,19)(2,9)(4,21,23,17,22,15,5,24)(10,26)(11,27)(12,14)(13,28)(20,32)(25,30)(29,31);(2,10)(4,22)(5,23)(9,26)(13,25)(15,21)(17,24)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=ei,f^b=fi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219575, #72219576, #72219577, … |
| maximal normal subgroup of | #72219575, #72219576, #72219577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44826);