Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator, prime radical | C24 | C4×C8 |
| Frattini, cosocle | C2×#431 | C22 |
| socle | C22 | #1280 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×#431 ▷ C22×C4 ▷ C22 ▷ C1 | |
215 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 215 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) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,11)(4,15,16,8)(5,9)(6,10)(7,12,19,25)(13,23)(14,24)(17,28,29,21)(18,22)(20,26,30,32)(27,31)(33,34,36,41,37,42,47,55,38,40,46,52,48,53,59,62)(35,39,44,49,45,51,58,60,43,50,56,54,57,61,64,63);(1,3,5,13,6,14,18,27)(2,7,9,20,10,19,22,30)(4,12,17,26,16,25,29,32)(8,11,21,23,15,24,28,31)(33,35)(34,39,40,50)(36,44)(37,45)(38,43)(41,54,52,49)(42,51,53,61)(46,56)(47,58)(48,57)(55,63,62,60)(59,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,c^b=ci,d^b=df,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 178);