Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #3100 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,8,7)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,20,13,21,22,33,14,19,23,31,24,32,34,39)(10,15,16,26,17,27,28,37,18,25,29,35,30,36,38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eg,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72286610, C2×#483602, #72321994, … |
| maximal normal subgroup of | #72286610, C2×#483602, #72321994, … |
| maximal quotient of | #72286643, #72286931, #72287116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390798);