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 | C22×D16 | C22 |
| 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 | |
384 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 118 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)(9,10,12,16)(11,18,15,29)(13,17,23,30)(14,28,24,40)(19,37,31,25)(20,22,32,26)(21,33,27,35)(34,39,36,38);(2,5)(6,8)(10,16)(17,30)(18,29)(19,31)(20,32)(28,40)(33,35)(34,36);(10,17)(11,15)(13,25)(14,26)(16,30)(18,35)(19,20)(21,38)(22,24)(23,37)(27,39)(28,34)(29,33)(31,32)(36,40);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,18,16,29)(13,21,23,27)(14,22,24,26)(17,33,30,35)(19,36,31,34)(20,28,32,40)(25,39,37,38)Polycyclic presentation:
2^9|a^2=i,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,d^c=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234373, #72234374, #72234388, … |
| maximal normal subgroup of | #72234373, #72234374, #72234388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420200);