Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
287 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 110 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)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2)(3,10)(4,13)(5,7)(6,14)(8,12)(9,15)(11,16);(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=ch,d^b=dfh,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72287293, #72320321, #72320404, … |
| maximal normal subgroup of | #72287293, #72320321, #72320404, … |
| maximal quotient of | #72287158 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29661);