Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
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 | #358 |
| 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 | |
135 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,5,7,6,3,4,8)(9,10,13,19,16,23,37,22)(11,24,26,56,30,57,29,68)(12,20,32,51,34,54,69,53)(14,38,35,49,33,18,46,48)(15,40,36,21,39,47,31,52)(17,41,42,25,45,59,50,61)(27,63,62,70,60,55,65,44)(28,66,58,71,64,72,67,43);(1,3,6,2)(4,7,5,8)(9,11,14,27,39,64,69,59,37,26,46,62,36,67,34,61,16,30,33,60,15,28,32,41,13,29,35,65,31,58,12,25)(10,17,20,43,52,70,49,56,19,50,51,72,40,63,18,24,23,45,54,71,21,44,48,68,22,42,53,66,47,55,38,57)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=gi,e^2=fgi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1612);