Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1469 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20);(5,7)(6,10,12,13)(8,14)(9,15)(11,19,16,18)(17,20);(5,8)(6,11,12,16)(7,14)(9,17)(10,18,13,19)(15,20);(2,4)(14,20)(18,19)Polycyclic presentation:
2^9|b^2=g,c^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237852, #72237912, #72237914, … |
| maximal normal subgroup of | #72237852, #72237912, #72237914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386774);