Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
5498 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 5498 conjugacy classes of subgroups — too many to draw. Only its 370 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)(5,6,9,11,8,7,12,10)(13,14)(15,18)(16,19)(17,20);(2,4)(9,12)(10,11)(14,18)(15,16)(19,20);(5,7)(6,8)(9,10)(11,12)(14,19)(15,16)(18,20);(14,20)(18,19)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226686, #72240335, #72240337, … |
| maximal normal subgroup of | #72226686, #72240335, #72240337, … |
| maximal quotient of | #72219616, #72219618, #72224609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93886);