Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#97 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,12,17,13,19,24,32,14,18,23,33,25,16,29,31)(11,15,21,27,22,26,36,37,20,28,34,39,35,30,40,38);(1,2,3,6,4,5,7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,30)(17,27)(18,28)(19,26)(23,34)(24,36)(25,35)(29,40)(31,38)(32,37)(33,39);(2,5)(6,8)(10,16,18,19)(11,20)(12,23)(15,26,28,30)(17,32,33,31)(22,35)(24,29)(27,38,39,37)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,c^a=cg,f^a=fi,c^b=chi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224814, #72224823, #72224825, … |
| maximal normal subgroup of | #72224814, #72224823, #72224825, … |
| maximal quotient of | #72227019, #72227020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33741);