Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#334 |
| prime radical | C23 | #334 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
177 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 177 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:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15);(2,7)(6,12)(8,14)(13,15)Polycyclic presentation:
2^9|a^2=e,b^2=d,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231928, #72231931, #72231945, … |
| maximal normal subgroup of | #72231928, #72231931, #72231945, … |
| maximal quotient of | #72286585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30116);