Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,19,20,15)(12,17)(13,18)(14,16)(21,31,33,26)(22,32,34,27)(23,30)(24,28)(25,29)(35,39,40,37)(36,38);(9,11,12,21,13,22,23,35,14,20,24,33,25,34,36,40)(10,15,17,26,18,27,30,37,16,19,28,31,29,32,38,39);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=di,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72319915, #72319950, #72319959, … |
| maximal normal subgroup of | #72319915, #72319950, #72319959, … |
| maximal quotient of | C2×#126568 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33764);