Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137580 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9064 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,12,21,27,5,14,4,11,8,10,20,28)(3,15,16,24,17,29,19,25,18,31,32,26,23,30,22,9);(1,3,5,18)(2,9)(4,19,6,22)(7,23,8,17)(10,24)(11,29)(12,26)(13,30)(14,25)(15,28)(16,21,32,20)(27,31);(2,10)(4,20)(6,21)(7,8)(9,24)(12,14)(15,31)(16,19)(17,23)(22,32)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=g,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219574, #72219597, #72219600, … |
| maximal normal subgroup of | #72219574, #72219597, #72219600, … |
| maximal quotient of | #72226841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44776);