Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132079 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8459 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | #62387 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 32 points:
(1,2,5,11,4,10,19,29,6,12,21,31,18,8,20,26)(3,13,15,32,16,7,23,25,17,24,22,30,14,9,27,28);(2,7)(4,18)(5,20)(8,24)(9,12)(10,13)(11,30)(14,16)(15,27)(19,21)(22,23)(25,29)(26,28)(31,32);(2,8,12,10)(5,21)(7,24,9,13)(11,29,31,26)(15,22)(19,20)(23,27)(25,32,28,30)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39275);