Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | C2×#379 | C22 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
849 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 849 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,14)(15,20)(16,18)(17,19);(5,7)(6,10,12,18)(8,16)(9,15)(11,13)(14,17)(19,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,b^a=bc,c^a=cf,e^a=efgi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72238058, #72267776, #72285097, … |
| maximal normal subgroup of | #72238058, #72267776, #72285097, … |
| maximal quotient of | #72238067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1487);