Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9078 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,10,6,12,4,20,7,13,21,9,15,11,22,27)(3,14,18,25,19,29,17,8,16,24,32,26,30,28,31,23);(1,3,15,30,7,16,6,19)(2,8,13,23)(4,17,5,18,22,31,21,32)(9,14,10,24)(11,25,12,26)(20,28,27,29);(3,16)(17,31)(18,32)(19,30)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,g^a=gi,c^b=cgh,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219590, #72219597, #72219599, … |
| maximal normal subgroup of | #72219590, #72219597, #72219599, … |
| maximal quotient of | C2×#137663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44859);