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 | |
507 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 507 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:
(1,2,7,25)(3,13,26,11)(4,18,24,19)(5,10,6,12)(8,17,16,15)(9,28,22,29)(14,30,21,31)(20,27,23,32);(1,3)(2,8)(4,19,23,27)(5,15)(6,17)(7,26)(9,29,31,21)(10,13)(11,12)(14,22,28,30)(16,25)(18,20,32,24);(1,4)(2,9)(3,14)(5,24)(6,23)(7,20)(8,27)(10,22)(11,19)(12,31)(13,32)(15,21)(16,18)(17,28)(25,30)(26,29)Polycyclic presentation:
2^9|a^2=dg,b^2=di,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 | #72260645, #72260652, #72260654, … |
| maximal normal subgroup of | #72260645, #72260652, #72260654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51843);