Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
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 78 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,23,25,51)(12,19)(13,33)(14,36)(15,39)(16,22)(17,24,40,54)(18,30)(20,38)(21,37)(26,59,55,50)(27,61,57,72)(28,63,58,52)(29,46)(31,66)(32,49)(34,44)(35,45)(41,53,68,62)(42,70,69,60)(43,71,67,56)(47,65)(48,64);(2,5)(3,8)(6,7)(9,11,13,26,34,57,64,71,37,58,65,70,38,62,32,54,16,25,30,55,14,27,31,56,15,28,35,60,29,53,12,24)(10,17,19,41,46,69,45,52,39,67,66,72,36,59,18,23,22,40,49,68,20,42,47,63,21,43,48,61,44,50,33,51)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,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, 1552);