Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator, Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
767 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 767 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,5)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,23)(19,25)(21,24,29,31)(22,30)(26,27)(28,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26,28,22)(20,23)(21,24)(25,27,32,30)(29,31);(2,6)(4,8)(5,7)(13,16)(17,19)(18,22)(20,28)(21,30)(23,26)(24,32)(25,31)(27,29)Polycyclic presentation:
2^9|a^2=d,b^2=dgh,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fg,d^c=dg,e^c=egh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72266591 |
| maximal normal subgroup of | #72266591 |
| maximal quotient of | #72266623 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11965);