Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513173 |
| 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 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,15,20)(11,16,14,29)(12,17,26,34)(13,27,28,40)(18,35,36,23)(19,22,37,24)(21,32,25,30)(31,38,33,39);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,20,29)(12,21,26,25)(13,22,28,24)(17,32,34,30)(18,31,36,33)(19,27,37,40)(23,39,35,38);(1,3,5,4)(2,7,6,8)(10,17)(11,14)(12,23)(13,24)(16,30)(18,19)(20,34)(21,38)(22,28)(25,39)(26,35)(27,33)(29,32)(31,40)(36,37);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=e,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72234321, #72234341, #72322373, … |
| maximal normal subgroup of | #72234321, #72234341, #72322373, … |
| maximal quotient of | #72322876 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420496);