Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93290 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1274 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C4×#117 | C22 |
| socle | C2 | #7017 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C4×#117 ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 319 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)(3,13,14,30,17,25,31,27)(5,12,20,28)(6,9,21,26)(7,11,19,29)(8,22,23,16,24,18,32,15);(1,3)(2,8,11,24)(4,18,19,22)(5,15)(6,16)(7,17)(9,13,12,25)(10,27)(14,21,31,20)(23,26)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,f^2=i,g^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 | #72211060, #72211068, #72211075 |
| maximal normal subgroup of | #72211060, #72211068, #72211075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 486);