Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483009 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#329 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | #3498 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #2875 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
818 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,7,17,20,6,11,16,25,18,9,15,24)(3,8,13,23,14,19,29,30,12,21,26,31,27,22,28,32);(2,7)(4,15)(5,18)(8,19)(9,11)(10,25)(13,28)(14,27)(16,17)(21,22)(23,31)(26,29);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,21)(15,28)(16,26)(17,29)(18,27)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,22)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=f,e^2=h,f^2=g,g^2=i,b^a=be,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^e=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390205);