Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1846 |
| prime radical | C23 | #427 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,30,34,54)(20,36,39,66)(21,41)(22,28)(23,45)(24,29)(25,32,48,61)(26,38,50,40)(27,43)(31,55)(33,53,35,63)(37,67,69,49)(42,51)(44,71)(46,73)(47,60,72,62)(52,58)(56,77)(57,75)(59,76)(64,79)(65,68,80,70)(74,78);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,32,23,35,43,62,24,34,44,61,22,33,42,60)(18,25,27,30,28,47,41,53,29,48,51,54,45,72,71,63)(20,31,37,56,40,58,70,78,39,59,69,79,38,57,68,73)(26,46,49,52,36,64,65,55,50,74,67,75,66,77,80,76);(3,9)(6,12)(10,15)(13,16)(17,20)(18,26,29,50)(19,31)(21,37)(22,38)(23,40)(24,39)(25,46,48,74)(27,49,51,67)(28,36,45,66)(30,52,54,75)(32,56)(33,57)(34,59)(35,58)(41,65,71,80)(42,68)(43,70)(44,69)(47,64,72,77)(53,55,63,76)(60,73)(61,79)(62,78)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42949);