Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1365 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7274 |
| prime radical | C23 | #362 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,9,15,8,35)(3,16,18,42,22,51,21,37)(4,23,26,58,6,30,29,60)(7,32,25,48,34,43,27,14)(10,24,36,17,40,19,39,54)(11,41,44,31,13,46,45,28)(20,57,55,50,33,61,52,64)(38,53,49,56,47,62,63,59);(1,3,6,19,25,55,46,62,8,18,26,54,7,33,31,59,9,22,4,24,27,52,41,53,5,21,29,17,34,20,28,56)(2,10,13,37,43,63,58,64,12,39,45,51,14,47,30,61,15,40,11,42,32,49,60,50,35,36,44,16,48,38,23,57)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,e^2=fgi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225338, #72225340, #72225342, … |
| maximal normal subgroup of | #72225338, #72225340, #72225342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1562);