Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146086 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #482 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(30,32)(33,34)(35,45,50,40)(36,51,56,69)(37,43)(38,44)(39,42)(41,53,67,55)(46,77)(47,65,49,63)(48,76,84,64)(52,88)(54,87,78,74)(57,72)(58,70)(59,92,81,73)(60,68,90,93)(61,71)(62,79)(66,83)(75,85)(80,94)(82,95)(86,89)(91,96);(1,3)(2,7)(4,15)(5,18)(6,14)(8,20)(9,24)(10,21)(11,25)(12,17)(13,19)(16,28)(22,27)(23,31)(26,32)(29,30)(33,35,39,50)(34,40,42,45)(36,52,56,89)(37,47,58,49)(38,59,61,81)(41,66,67,96)(43,63,70,65)(44,73,71,92)(46,78,80,54)(48,82,84,57)(51,86,69,88)(53,83,55,91)(60,85,90,79)(62,93,75,68)(64,95,76,72)(74,77,87,94);(1,4,5,16,6,17,19,30)(2,8,9,22,10,23,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27)(33,36)(34,41,42,67)(35,46)(37,53)(38,60)(39,56)(40,62,45,75)(43,51,70,69)(44,74,71,87)(47,79)(48,83)(49,85)(50,80)(52,59)(54,57)(55,58)(61,90)(63,94,65,77)(64,86,76,88)(66,92,96,73)(68,95,93,72)(78,82)(81,89)(84,91)Polycyclic presentation:
2^9|a^2=ehi,b^2=i,c^2=d,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,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, 53285);