Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97058 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2049 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 2049 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(9,12)(13,14)(15,19,20,18)(16,17);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16,17,14)(15,19,20,18)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225675, #72266443, #72266448, … |
| maximal normal subgroup of | #72219238, #72225675, #72266443, … |
| maximal quotient of | #72219230, #72314163, #72314171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22545);