Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C23 | #408 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 80 points:
(1,2)(3,11,13,15)(4,14,7,8)(5,9)(6,10)(12,16)(17,18,22,29,24,30,45,59,23,27,44,46,21,28,41,56)(19,25,34,49,33,48,65,75,32,50,42,57,39,54,70,77)(20,35,38,68,37,53,69,58,36,55,43,71,40,26,51,52)(31,60,62,78,64,47,72,74,63,67,66,80,61,73,79,76);(1,3,5,14,6,13,16,8)(2,7,9,15,10,4,12,11)(17,19,22,42,24,33,45,70,23,32,44,34,21,39,41,65)(18,25,46,49,30,48,56,75,27,50,29,57,28,54,59,77)(20,31,43,62,37,64,51,72,36,63,38,66,40,61,69,79)(26,47,52,76,35,67,68,78,53,73,58,74,55,60,71,80);(1,4,6,7)(2,8,10,14)(3,12,13,9)(5,11,16,15)(17,20,23,36)(18,26,27,53)(19,31,32,63)(21,40,24,37)(22,43,44,38)(25,47,50,73)(28,55,30,35)(29,58,46,52)(33,64,39,61)(34,66,42,62)(41,69,45,51)(48,67,54,60)(49,76,57,74)(56,68,59,71)(65,79,70,72)(75,78,77,80)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=ei,f^c=fg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43980);