Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12759 |
| prime radical | #61105 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,6,13,8,15,26,30)(3,4,17,21,18,22,31,32)(5,12,23,11,24,29,7,14)(9,25,28,27,19,16,20,10);(1,3)(2,9)(4,14)(5,17)(6,25)(7,27)(8,18)(10,23)(11,22)(12,20)(13,21)(15,19)(16,26)(24,31)(28,29)(30,32);(1,4)(2,10)(3,14)(5,20)(6,21)(7,19)(8,22)(9,23)(11,18)(12,17)(13,25)(15,27)(16,30)(24,28)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=egh,f^b=fgh,g^b=gi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59383);