Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#493 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
636 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(18,22)(19,23)(20,26)(24,32)(25,30)(27,31)(28,29);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(17,18,23,29,20,24,27,30,21,25,31,32,26,28,19,22);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72216236, #72216241, #72216244, … |
| maximal normal subgroup of | #72216236, #72216241, #72216244, … |
| maximal quotient of | #72216261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398681);