Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,13,18,14,17,27,32,15,19,28,33,12,25,26,38)(11,20,23,35,24,30,37,39,21,34,36,40,22,16,29,31);(1,2,3,5,4,6,7,8)(9,11)(10,16,19,30)(12,22)(13,23)(14,24)(15,21)(17,20,25,34)(18,31,33,39)(26,29)(27,37)(28,36)(32,35,38,40);(11,21)(16,30)(20,34)(22,24)(23,36)(29,37)(31,39)(35,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=di,f^a=fi,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72224829, #72224834, #72224835, … |
| maximal normal subgroup of | #72224829, #72224834, #72224835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33749);