Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500306 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3227 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #24061 |
| prime radical | #60676 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 789 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 4 generators on 32 points:
(1,2)(3,12)(4,17)(5,10)(6,22)(7,11)(8,15)(9,21)(13,29)(14,28)(16,26)(18,32)(19,23)(20,25)(24,30)(27,31);(3,13,14,16)(4,18)(8,23,24,27)(9,25)(12,26,28,29)(15,31,30,19)(17,32)(20,21);(1,3,6,13,7,14,5,16)(2,8,10,27,11,24,22,23)(4,19,21,30,18,31,20,15)(9,26,17,28,25,29,32,12);(4,18)(9,25)(12,28)(15,30)(17,32)(19,31)(20,21)(26,29)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219593, #72226636, #72226637, … |
| maximal normal subgroup of | #72219593, #72226636, #72226637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407502);