Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139080 |
| 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)(33,34,38,44,63,66,60,43,39,45,61,77,40,46,37,59)(35,47,50,41,64,68,49,78,51,81,88,69,52,75,62,70)(36,53,56,73,65,76,54,74,57,71,91,42,58,72,55,79)(48,67,84,80,89,96,83,92,85,94,87,90,86,95,82,93);(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)(33,35,40,64,39,51,63,52)(34,41,66,70,45,69,46,78)(36,48,65,86,57,85,58,89)(37,50,61,49,60,88,38,62)(42,67,74,96,73,94,79,95)(43,75,77,81,59,68,44,47)(53,90,72,93,71,80,76,92)(54,87,91,83,55,84,56,82);(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)(33,36,39,57)(34,42,45,73)(35,48,51,85)(37,56,60,91)(38,55,61,54)(40,65,63,58)(41,67,69,94)(43,76,59,72)(44,71,77,53)(46,79,66,74)(47,80,81,90)(49,87,62,84)(50,82,88,83)(52,89,64,86)(68,93,75,92)(70,96,78,95)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=eg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=ehi,f^c=fghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46332);