Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26,31,34,32,36,40,35)(27,37,33,30,39,38,28,29);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,26,31,28,38,35,32,39,29,36,40,33,37,34);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,34,31,39,37,26,32,33,30,35,40,27,38,36)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=chi,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343501, #72343502, #72343503, … |
| maximal normal subgroup of | #72343501, #72343502, #72343503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42868);