Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417493 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35390 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3417 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 3417 conjugacy classes of subgroups — too many to draw. Only its 226 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 — 4 generators on 20 points:
(1,2)(3,4)(6,9)(10,12)(13,14)(15,17)(16,18)(19,20);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,19)(18,20);(5,7,8,11)(6,10,9,12)(17,20);(2,4)(6,9)(10,12)Polycyclic presentation:
2^9|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72317543, #72317551, #72317574 |
| maximal normal subgroup of | #72317543, #72317551, #72317574 |
| maximal quotient of | #72318316 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325324);