Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1804 |
| prime radical | C23 | #416 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
277 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 277 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,5,8,6,3,4,7)(9,10)(11,21,24,16)(12,26)(13,30)(14,36)(15,18)(17,28)(19,34)(20,33)(22,49,50,68)(23,53,31,64)(25,40,56,48)(27,60)(29,63)(32,46)(35,59)(37,45)(38,51,47,62)(39,57,44,66)(41,67)(42,61)(43,54)(52,70,58,72)(55,69,65,71);(9,11,14,25,15,24,33,56)(10,16,36,48,18,21,20,40)(12,22,37,58,27,50,29,52)(13,31,34,66,35,23,54,57)(17,38,41,69,42,47,46,71)(19,44,59,64,43,39,30,53)(26,49,45,72,60,68,63,70)(28,62,67,55,61,51,32,65);(1,3,6,2)(4,7,5,8)(9,12,15,27)(10,17,18,42)(11,22,24,50)(13,32,35,67)(14,37,33,29)(16,38,21,47)(19,45,43,63)(20,46,36,41)(23,55,31,65)(25,58,56,52)(26,59,60,30)(28,54,61,34)(39,70,44,72)(40,71,48,69)(49,64,68,53)(51,66,62,57)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=gh,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41831);