Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,20,16,22,15,32)(11,23,26,35,29,38,28,33)(12,18,30,37,31,21,14,19)(17,27,34,24,36,40,39,25);(2,5,7,6)(9,11,14,27,15,28,31,25,16,29,30,40,13,26,12,24)(10,17,21,35,20,34,19,38,22,36,18,33,32,39,37,23);(9,12,13,30,16,31,15,14)(10,18,20,37,22,21,32,19)(11,24,26,40,29,25,28,27)(17,33,34,23,36,35,39,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216219, #72216233, #72216234, … |
| maximal normal subgroup of | #72216219, #72216233, #72216234, … |
| maximal quotient of | C8×#1333, C2×#128537, #72225319 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35733);