Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152678 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #23906 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 809 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6,11,15)(4,9,13,7)(5,8)(10,14,12,16);(1,3,5,12)(2,6,9,15,8,16,7,14)(4,10,13,11);(1,4,5,13)(2,7,8,9)(3,10,12,11)(6,14,16,15)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213519, #72216567, #72216600, … |
| maximal normal subgroup of | #72213519, #72216567, #72216600, … |
| maximal quotient of | #72250643, C2×#152679, #72279404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59875);