Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,8,5,7,3,6)(9,10,13,19,15,17,32,21)(11,22,24,36,27,37,26,40)(12,28,30,18,31,20,14,33)(16,25,35,23,34,39,38,29);(2,6,7,8)(9,11,14,25,32,24,31,23,15,27,30,39,13,26,12,29)(10,16,20,36,19,38,33,22,17,34,28,40,21,35,18,37);(2,7)(6,8)(10,17)(16,34)(18,33)(19,21)(20,28)(22,37)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,d^b=di,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72319942, #72319964, #72319991, … |
| maximal normal subgroup of | #72319942, #72319964, #72319991, … |
| maximal quotient of | C2×#126866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34062);