Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| 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:
(1,2,5,11,6,13,4,20,7,9,22,10,15,12,21,27)(3,14,18,25,19,29,16,8,17,24,32,26,30,28,31,23);(1,3,15,30,7,17,6,19)(2,8,9,23)(4,16,5,18,21,31,22,32)(10,14,11,24)(12,25,13,26)(20,28,27,29);(2,9)(4,21)(5,22)(8,23)(12,13)(16,31)(18,32)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,e^b=ei,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219574, #72219576, #72219577, … |
| maximal normal subgroup of | #72219574, #72219576, #72219577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44825);