Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1464 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1464 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,15,19,13,18)(11,16,24,32,23,33,22,31)(12,25,29,17,27,35,28,34)(21,36,38,30,26,39,37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,26,27,21)(13,22)(14,24)(15,23)(17,30,34,40)(18,31)(19,33)(20,32)(25,36,35,39)(28,38,29,37);(5,8)(9,12)(10,17)(11,21)(13,29)(14,28)(15,27)(16,30)(18,35)(19,34)(20,25)(22,38)(23,26)(24,37)(31,39)(32,36)(33,40)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266588 |
| maximal normal subgroup of | #72266588 |
| maximal quotient of | #72266592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23559);