Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129617 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #37129 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1259 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 138 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)(7,14,16,11)(8,12,18,19)(9,13)(10,15)(17,20);(1,2)(3,4)(5,7)(6,10)(8,15)(9,16)(11,19)(12,14)(13,20)(17,18);(2,4)(5,7)(6,11)(8,17)(9,16)(10,19)(12,20)(13,14)(15,18)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225802, #72233038, #72233087, … |
| maximal normal subgroup of | #72225802, #72233038, #72233087, … |
| maximal quotient of | #72225800, #72225804, #72233022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54979);