Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123363 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1667 |
| commutator, Frattini, cosocle | C82 | C23 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
197 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 3 generators on 32 points:
(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6);(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)Polycyclic presentation:
2^9|b^2=df,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,e^c=eg,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72225448, C2×#123363, D162, … |
| maximal normal subgroup of | #72225448, C2×#123363, D162, … |
| maximal quotient of | #72225437, #72225580, #72225581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30559);