Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1970 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9663 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 311 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:
(1,2,6,9,7,14,5,24,8,15,4,12,22,13,23,11)(3,16,18,27,17,30,19,10,20,25,21,28,31,26,32,29);(2,9,13,11,15,12,14,24)(3,17,20,31)(4,6)(5,23)(10,26,28,16,29,30,27,25)(18,32,21,19);(1,3,4,21,7,17,23,32,8,20,6,18,22,31,5,19)(2,10,24,26,13,27,9,25,15,29,11,30,14,28,12,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224956, #72224959, #72224966, … |
| maximal normal subgroup of | #72224956, #72224959, #72224966, … |
| maximal quotient of | C2×#140776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47972);