Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23173 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 32 points:
(1,2,6,14,7,13,23,28,8,15,22,10,5,12,4,11)(3,16,18,29,19,27,17,31,21,30,24,32,20,9,25,26);(1,3,4,17,5,20,22,18,8,21,23,25,7,19,6,24)(2,9,11,29,12,30,10,26,15,27,28,32,13,16,14,31);(1,4,5,22,8,23,7,6)(2,10,13,11,15,14,12,28)(3,17,20,18,21,25,19,24)(9,26,16,29,27,31,30,32);(1,4,5,22,8,23,7,6)(2,11,12,10,15,28,13,14)(3,18,19,17,21,24,20,25)(9,26,16,29,27,31,30,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72225428, #72225431, #72225458, … |
| maximal normal subgroup of | #72225428, #72225431, #72225458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405746);