Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #56344 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
808 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 808 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,4,10,20,8,23,24)(3,13,16,27,31,9,25,26)(5,11,17,29,6,12,19,30)(7,18,22,14,21,32,28,15);(2,7)(3,14)(8,21)(9,11)(10,27)(12,13)(15,31)(16,32)(18,25)(22,30)(24,26)(28,29);(2,8)(3,15)(4,17)(7,21)(9,13)(11,12)(14,31)(16,25)(18,32)(19,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fg,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72231818 |
| maximal normal subgroup of | #72231818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40089);