Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12263 |
| commutator, Frattini, cosocle | #379 | C23 |
| prime radical | #2021 | C22 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,2,6,10)(3,11,14,25)(4,15)(5,9,18,23)(7,19,21,13)(8,17)(12,24)(16,22)(20,28)(26,29)(27,32)(30,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,29)(16,28)(17,27)(18,19)(22,24)(23,25)(26,31)(30,32);(1,4)(2,8)(3,12)(5,16,18,31)(6,17)(7,20)(9,22,23,30)(10,15)(11,24,25,32)(13,28,19,26)(14,27)(21,29)Polycyclic presentation:
2^9|a^2=fi,c^2=fgh,d^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ehi,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72284603, #72284679, #72284711, … |
| maximal normal subgroup of | #72284603, #72284679, #72284711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58659);