Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9600 |
| prime radical | C23 | #443 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,13,7,11,24,26,8,14,25,31,5,23,20,12)(3,10,16,27,15,9,22,28,18,29,21,32,17,30,4,19);(1,3,6,16,7,15,24,22,8,18,25,21,5,17,20,4)(2,9,13,28,11,29,26,32,14,30,31,19,23,10,12,27);(1,4,20,17,5,21,25,18,8,22,24,15,7,16,6,3)(2,10,13,27,11,9,26,28,14,29,31,32,23,30,12,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,d^2=i,e^2=gi,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=cgh,d^b=di,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320359, #72320727, #72320764 |
| maximal normal subgroup of | #72320359, #72320727, #72320764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47568);