Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #13661 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,27)(14,29)(15,20)(18,23)(19,24)(21,30)(22,36)(25,33)(26,32)(28,40)(31,37)(34,38)(35,39);(2,5)(6,8);(10,16)(11,14)(12,23)(13,24)(17,33)(18,19)(20,32)(21,37)(22,28)(25,38)(26,39)(27,31)(29,30)(34,40)(35,36);(1,3,4,7)(2,6,5,8)(9,11,15,14)(10,17,20,29)(12,21,26,25)(13,22,28,24)(16,30,32,33)(18,34,35,31)(19,27,36,40)(23,38,39,37)Polycyclic presentation:
2^9|d^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^c=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232028, #72234321, #72234324, … |
| maximal normal subgroup of | #72232028, #72234321, #72234324, … |
| maximal quotient of | #72286704, #72322606 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420204);