Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #466 | C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
259 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,6,7,15)(3,9)(4,12)(8,10,13,16)(11,14)(17,18,21,25,22,26,32,31,23,27,20,29,30,19,28,24);(1,2)(3,10)(4,13)(5,7)(6,14)(8,12)(9,15)(11,16)(17,19)(18,22)(20,24)(21,31)(23,26)(25,28)(27,30)(29,32);(2,7)(6,15)(8,13)(10,16)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=dfh,e^b=eh,f^b=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286956, #72286988, #72287052, … |
| maximal normal subgroup of | #72286956, #72286988, #72287052, … |
| maximal quotient of | #72287158, #72323197 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29667);