Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3296 conjugacy classes of subgroups; only the 386 normal subgroups are shown.
This group has 3296 conjugacy classes of subgroups — too many to draw. Only its 386 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,20)(18,19);(2,4)(6,9)(10,12)(13,14)(15,17)(16,18)(19,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,d^2=g,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261515, #72278420, #72278500, … |
| maximal normal subgroup of | #72261515, #72278420, #72278500, … |
| maximal quotient of | #72261587, #72281629, #72281779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266059);