Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144613 |
| 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×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 565 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:
(1,2,7,11)(3,12)(4,8,20,25)(5,21,6,22)(9,26)(10,19,31,18)(13,24)(14,30)(15,29)(16,32)(17,28)(23,27);(2,8)(3,13,16,23)(9,12,29,32)(10,21)(11,25)(14,15,17,26)(22,31)(24,28,27,30);(1,3)(2,9)(4,17)(5,13)(6,23)(7,16)(8,24)(10,30)(11,29)(12,22)(14,20)(15,19)(18,26)(21,32)(25,27)(28,31)Polycyclic presentation:
2^9|a^2=dgi,b^2=di,e^2=gi,f^2=g,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,d^c=dg,e^c=egi,f^c=fg,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233366, #72233487, #72233493, … |
| maximal normal subgroup of | #72233366, #72233487, #72233493, … |
| maximal quotient of | #72321528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51815);