Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139074 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #449 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,15,27,14,8)(4,17,19,25,20,9,18,24)(13,26,30,22,29,32,28,31)(33,34,38,45,63,66,61,43,39,46,62,77,40,47,37,44)(35,48,52,41,64,68,50,78,53,81,88,69,54,75,51,70)(36,55,57,73,65,76,56,74,58,71,60,42,59,72,90,79)(49,67,84,92,89,95,83,91,85,93,87,80,86,94,82,96);(1,3)(2,8,11,23)(4,13)(5,14)(6,16)(7,15)(9,22,17,31)(10,12,21,27)(18,28)(19,30)(20,29)(24,32,25,26)(33,35,40,64,39,53,63,54)(34,41,66,70,46,69,47,78)(36,49,65,86,58,85,59,89)(37,52,62,50,61,88,38,51)(42,67,74,95,73,93,79,94)(43,75,77,81,44,68,45,48)(55,80,72,96,71,92,76,91)(56,87,60,83,90,84,57,82);(1,4,7,20)(2,9,11,17)(3,13,15,29)(5,19,6,18)(8,22,23,31)(10,25,21,24)(12,26,27,32)(14,30,16,28)(33,36,39,58)(34,42,46,73)(35,49,53,85)(37,60,61,57)(38,56,62,90)(40,65,63,59)(41,67,69,93)(43,72,44,76)(45,55,77,71)(47,79,66,74)(48,80,81,92)(50,84,51,87)(52,83,88,82)(54,89,64,86)(68,91,75,96)(70,95,78,94)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=ehi,f^c=fgh,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46326);