Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150171 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | C2×#1352 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,12,15,29)(4,18)(5,11)(6,8)(7,14)(9,20)(10,13)(16,30,17,26)(19,24)(21,25)(22,28)(23,31)(27,32);(1,3,13,16,6,15,5,17)(2,7,11,24,8,23,10,27)(4,19,21,31,22,32,20,14)(9,26,28,12,25,30,18,29);(2,8)(7,23)(9,25)(10,11)(12,29)(18,28)(24,27)(26,30)Polycyclic presentation:
2^9|a^2=e,b^2=fh,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72339753, #72339758, #72339759 |
| maximal normal subgroup of | #72339753, #72339758, #72339759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57475);