Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13377 |
| prime radical | #3481 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,4,15,8,12,7,11)(3,5,10,6,14,16,13,9);(1,3,4,10,8,14,7,13)(2,9,15,5,12,6,11,16);(2,10)(3,11)(12,13)(14,15)Polycyclic presentation:
2^9|a^2=dh,b^2=dh,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72338939, #72339673 |
| maximal normal subgroup of | #72338939, #72339673 |
| maximal quotient of | #72338941, #72339054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60424);