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 | |
828 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,8)(4,6)(5,7)(10,11,15,16);(1,3,4,2,5,6,8,7)(9,10,13,11,14,15,12,16);(9,11,12,10,14,16,13,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,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 | #1042, #72226268, #72226270, … |
| maximal normal subgroup of | #1042, #72226268, #72226270, … |
| maximal quotient of | C2×#110176, #72239511, #72239512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17372);