Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1380 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,5,7,6,3,4,8)(10,16,17,22)(11,21)(12,24)(13,28)(15,29)(18,30,32,37)(19,36,33,40)(20,35,34,31)(23,26)(25,27)(38,39);(1,3,6,2)(4,7,5,8)(9,10,12,18,26,33,38,31,29,34,39,40,28,37,24,16,14,17,21,32,13,19,25,35,15,20,27,36,23,30,11,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=gi,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=cg,d^b=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225486, #72225497, #72225505 |
| maximal normal subgroup of | #72225486, #72225497, #72225505 |
| maximal quotient of | #72225484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1611);