Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11460 |
| prime radical | C24 | M32 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 226 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:
(1,2,3,9,4,10,13,24,5,11,14,23,15,25,27,32,6,7,12,18,16,19,26,29,17,8,20,21,28,22,30,31);(2,7)(8,11)(10,19)(22,25);(2,8,7,11)(3,12)(9,23,18,21)(10,22,19,25)(13,26)(14,20)(24,32,29,31)(27,30)Polycyclic presentation:
2^9|a^2=e,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,e^b=ef,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287351, #72287365, #72287376, … |
| maximal normal subgroup of | #72287351, #72287365, #72287376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56855);