Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
684 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,10,15)(4,11,12,14)(5,9)(6,8)(13,16)(17,18,21,27,23,28,22,26)(19,29,30,32,31,20,25,24);(1,3)(2,7)(4,8)(5,10)(6,14)(9,15)(11,13)(12,16)(17,19,22,25,23,31,21,30)(18,24,26,20,28,32,27,29);(1,4,6,15,5,12,13,7)(2,3,8,14,9,10,16,11)(17,20,21,24,23,29,22,32)(18,25,27,19,28,30,26,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=ehi,f^c=fhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72209984, #72240066, #72240067, … |
| maximal normal subgroup of | #72209984, #72240066, #72240067, … |
| maximal quotient of | #72238287, #72238831, #72238849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17375);