Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140362 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1941 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9597 |
| prime radical | C23 | #444 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,7,13,5,14,26,28,8,15,6,27,22,11,25,12)(3,16,18,29,19,10,23,32,20,31,4,21,17,9,24,30);(1,3,7,18,5,19,26,23,8,20,6,4,22,17,25,24)(2,9,13,30,14,16,28,29,15,10,27,32,11,31,12,21);(1,4,6,17,5,24,25,3,8,18,7,19,22,23,26,20)(2,10,12,21,11,16,27,32,15,9,28,29,14,31,13,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224951, #72224959, #72224966, … |
| maximal normal subgroup of | #72224951, #72224959, #72224966, … |
| maximal quotient of | C2×#140362 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47558);