Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #93508 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1289 |
| prime radical | C24 | #107 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18,21,27,24,30,45,59)(19,31,33,62,36,63,43,74)(20,37,39,70,40,57,44,75)(22,28,46,60,23,29,47,61)(25,48,49,65,52,67,76,32)(26,53,55,41,56,73,58,42)(34,64,68,80,35,54,69,79)(38,51,71,78,66,50,72,77);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,25)(20,38,40,66)(21,43,45,33)(22,34)(23,35)(24,36)(26,54,56,64)(27,49,59,76)(28,50)(29,51)(30,52)(31,57,63,37)(32,42,65,41)(39,72)(44,71)(46,69,47,68)(48,53)(55,79)(58,80)(60,77,61,78)(62,70)(67,73)(74,75)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 819);