Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1316 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | #2112 | C22 |
| socle | C2 | #7054 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2112 ◁ G | |
| LECS | G ▷ #2112 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 32 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:
(1,2,5,11,6,13,24,38,7,10,25,34,27,35,53,58,8,12,23,39,28,40,51,61,4,19,21,37,22,47,50,60)(3,14,16,42,17,43,46,63,15,9,29,30,44,32,55,57,18,33,45,56,26,54,52,64,20,31,48,36,49,41,62,59);(1,3)(2,9)(4,20)(5,16)(6,26)(7,15)(8,18)(10,33)(11,36)(12,31)(13,41)(14,19)(17,28)(21,48)(22,44)(23,45)(24,52)(25,29)(27,49)(30,39)(32,40)(34,42)(35,43)(37,56)(38,57)(46,51)(47,54)(50,55)(53,62)(58,64)(59,61)(60,63)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=dei,f^b=fi,d^c=dg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 872);