Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | C2×C22⋊C16 | C22 |
| socle | C22 | #1314 |
| prime radical | C24 | M32 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C2×C22⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 80 points:
(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18,21,27,22,28,40,53,23,29,41,54,43,56,68,76,24,30,42,55,44,57,69,77,20,26,37,50,38,51,65,74)(19,31,33,58,34,59,62,78,32,48,60,52,61,72,79,75,35,49,39,66,63,73,67,80,36,25,45,46,64,47,70,71);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)(17,19)(18,25)(20,36)(21,39)(22,34)(23,32)(24,35)(26,49)(27,52)(28,47)(29,31)(30,48)(33,42)(37,60)(38,64)(40,67)(41,45)(43,61)(44,63)(46,55)(50,58)(51,73)(53,75)(54,66)(56,59)(57,72)(62,69)(65,79)(68,70)(71,77)(74,78)(76,80)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=i,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=ci,d^b=dei,d^c=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 878);