Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| 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 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
526 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)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,19,26,20,27,31,37,21,25,32,35,33,36,39,40);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,19,13,20,22,31,14,21,23,32,24,33,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(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,d^a=di,e^a=eg,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286610, C2×#483581, #72322004, … |
| maximal normal subgroup of | #72286610, C2×#483581, #72322004, … |
| maximal quotient of | #72286314, #72286956, #72287088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390777);