Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153235 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13377 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #3549 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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:
(1,2,8,11,7,14,4,13)(3,15,10,9,16,5,12,6);(1,3,8,10,7,16,4,12)(2,9,11,5,14,6,13,15);(2,10)(3,11)(4,8)(6,9)(12,14)(13,16)Polycyclic presentation:
2^9|a^2=dh,b^2=dh,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,d^c=di,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72338936, #72338939 |
| maximal normal subgroup of | #72338936, #72338939 |
| maximal quotient of | #72338937, #72338941, #72339054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60431);