Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149894 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2122 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #13218 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,7,12,9,15,25,31)(3,16,18,27,20,11,28,10)(4,21,22,26,19,30,17,32)(5,23,6,14,24,29,8,13);(1,3,6,4,9,20,8,19)(2,10)(5,18,25,17,24,28,7,22)(11,29)(12,21)(13,26)(14,32)(15,27)(16,23)(30,31);(1,4,6,20,9,19,8,3)(2,11,13,21,15,16,14,30)(5,18,7,22,24,28,25,17)(10,23,26,31,27,29,32,12)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60110);