Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144629 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,8)(3,11,13,27)(5,17)(6,15)(9,24,22,25)(10,18)(12,19,28,30)(14,20)(21,32,31,26)(23,29);(1,2,4,8)(3,12)(5,18,14,29)(6,16,15,7)(9,22)(10,17,23,20)(11,19)(13,28)(21,31)(24,25)(26,32)(27,30);(1,3)(2,9)(4,13)(5,19)(6,21)(7,12)(8,22)(10,26)(11,20)(14,30)(15,31)(16,28)(17,27)(18,25)(23,32)(24,29)Polycyclic presentation:
2^9|a^2=dg,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fghi,c^b=cf,e^b=eh,f^b=fgi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72260647, #72260652, #72260655, … |
| maximal normal subgroup of | #72260647, #72260652, #72260655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51842);