Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 541 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:
(2,7,9,23)(3,11)(4,16)(8,26,27,21)(10,25,24,12)(14,31)(15,22)(17,20)(18,28,29,30)(19,32);(2,8,9,27)(4,17)(7,21,23,26)(10,29,24,18)(12,30,25,28)(14,19)(16,20)(31,32);(1,2)(3,12)(4,18)(5,9)(6,10)(7,22)(8,16)(11,23)(13,24)(14,21)(15,25)(17,27)(19,28)(20,29)(26,31)(30,32)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=gi,f^2=gi,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,e^c=egi,f^c=fgi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72266619, #72266668 |
| maximal normal subgroup of | #72266619 |
| maximal quotient of | #72266617, #72266622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51806);