Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C22×2−1+4 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,22,33)(12,16)(13,25,28,17)(14,18)(15,21)(20,36,38,35)(23,34,27,31)(24,32)(26,30)(29,37)(39,40);(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,33)(18,30)(19,25)(21,28)(23,39)(24,38)(27,37)(31,35)(32,40)(34,36);(9,12,14,24)(10,16,18,32)(11,20,22,38)(13,27,28,23)(15,29,30,40)(17,34,25,31)(19,35,33,36)(21,39,26,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36419);