Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| 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 | #3169 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,8,7)(9,10,14,19,13,20,31,37,15,17,27,22,32,18,35,36)(11,21,25,38,26,34,40,12,23,29,39,28,24,16,33,30);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,26)(14,25)(15,23)(17,34)(18,29)(19,30)(20,21)(24,32)(27,39)(28,36)(31,40)(33,35)(37,38);(2,6)(5,7)(9,12,27,21,13,28,35,34,15,30,14,29,32,38,31,16)(10,11,22,39,20,26,36,33,17,23,19,25,18,24,37,40);(10,17)(11,23)(18,20)(19,22)(24,26)(25,39)(33,40)(36,37)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,d^a=di,e^a=eg,d^b=di,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225220, #72225221, #72286314, … |
| maximal normal subgroup of | #72225220, #72225221, #72286314, … |
| maximal quotient of | #72225221, #72286483, #72287193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390779);