Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,27)(11,20,22,16,23,31,28,30)(12,17,25,33,26,32,24,37)(21,34,35,29,36,40,38,39);(2,5)(6,8);(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,29)(18,30)(19,31)(20,27)(24,38)(25,35)(26,36)(32,39)(33,40)(34,37);(9,12,14,25,15,26,13,24)(10,17,19,33,18,32,27,37)(11,21,22,35,23,36,28,38)(16,29,31,40,30,39,20,34);(1,3,4,7)(2,6,5,8)(10,18)(16,30)(17,32)(19,27)(20,31)(29,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72285591, #72313507, #72315637, … |
| maximal normal subgroup of | #72285591, #72313507, #72315637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280556);