Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8404 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | #60937 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,10,6,12,22,24,7,11,4,17,19,30,20,32)(3,9,15,28,16,29,21,8,14,26,13,27,23,31,18,25);(1,3)(2,8)(4,18)(5,21)(6,23)(7,14)(9,24)(10,29)(11,25)(12,28)(13,20)(15,22)(16,19)(17,31)(26,32)(27,30);(1,3)(2,9,11,26)(4,13)(5,15)(6,16)(7,14)(8,24,25,32)(10,27,17,28)(12,29,30,31)(18,20)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72339831 |
| maximal normal subgroup of | #72339831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38632);