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 | #34981 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
2288 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 2288 conjugacy classes of subgroups — too many to draw. Only its 166 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)(17,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=ei,g^b=gh,d^c=di,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219236, C2×#111462, #72225626, … |
| maximal normal subgroup of | #72219236, #72219238, C2×#111462, … |
| maximal quotient of | #72219230, C2×#111462, #72235059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18658);