Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112870 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5,7,8)(4,6)(9,10,14,21,16,20,31,30,15,19,22,35,13,11,17,24)(12,25,28,39,29,34,38,40,26,23,36,37,27,18,32,33);(1,3,4,7)(2,5)(6,8)(9,11,22,24,16,10,17,21,15,20,14,30,13,19,31,35)(12,18,28,40,29,25,38,37,26,34,36,33,27,23,32,39);(2,6)(5,8)(9,12,16,29,15,26,13,27)(10,18,11,23,19,34,20,25)(14,28,31,38,22,36,17,32)(21,33,24,37,35,40,30,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fg,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47898);