Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7412 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C4×#124 | C22 |
| socle | C22 | #1393 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
261 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,20,22,36)(12,24)(13,18)(14,17)(15,21,30,37)(16,26)(19,28)(23,38,29,32)(25,31,39,35)(27,33)(34,40);(2,6,8,7)(9,11)(10,15,17,31)(12,25,26,39)(13,22)(14,29,28,23)(16,32,33,20)(18,30,19,35)(21,27)(24,38,34,36)(37,40)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=fhi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gh,h^b=hi,e^c=ehi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1576);