Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2418 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 2418 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(5,6)(7,11,12,9)(8,10)(13,14)(15,18,20,17)(16,19);(1,2)(3,4)(13,15)(14,17)(16,20)(18,19);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18,19,17)(16,20)Polycyclic presentation:
2^9|a^2=g,c^2=d,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,g^b=gh,e^c=eh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237498, #72237499, #72240425, … |
| maximal normal subgroup of | #72237498, #72237499, #72240425, … |
| maximal quotient of | #72240491, #72240496, #72251028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26690);