Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93290 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1274 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×C4×M16 | C22 |
| socle | C2 | #7017 |
| prime radical | C24 | #101 |
| LCS | G ▷ C42 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4×M16 ▷ C22×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 2 generators on 32 points:
(1,2,4,10,6,11,20,28,7,12,19,29,5,9,21,26)(3,13,14,30,15,25,18,32,17,24,31,27,16,8,22,23);(1,3,5,15,7,17,6,16)(2,8,11,24,12,25,9,13)(4,18,20,14,19,22,21,31)(10,27,26,23,29,30,28,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239217, #72239218, #72239220 |
| maximal normal subgroup of | #72239217, #72239218, #72239220 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489);