Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150413 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | #11786 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(2,8,10,24)(4,17,20,22)(9,23,26,30)(11,18,21,25)(12,27,28,32)(13,29,19,31);(1,2,5,21,7,10,6,11)(3,12,14,9,16,28,15,26)(4,18,17,24,20,25,22,8)(13,30,29,32,19,23,31,27);(1,3,6,15,7,16,5,14)(2,9,21,28,10,26,11,12)(4,19,17,31,20,13,22,29)(8,23,25,32,24,30,18,27)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=gh,d^2=ghi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57726);