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)(35,47,51,41)(36,52,56,76)(37,44)(38,45)(39,46)(40,43)(42,54,68,63)(48,79)(49,75,62,65)(50,78,85,66)(53,87)(55,73,80,96)(57,71)(58,90,91,72)(59,92,93,69)(60,74)(61,70)(64,81)(67,88)(77,82)(83,94)(84,95)(86,89);(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)(33,35,40,51)(34,41,43,47)(36,53,56,89)(37,49,39,62)(38,58,61,91)(42,67,68,95)(44,65,46,75)(45,72,70,90)(48,80,83,55)(50,57,85,60)(52,86,76,87)(54,88,63,84)(59,81,93,82)(64,92,77,69)(66,71,78,74)(73,94,96,79);(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)(34,42,43,68)(35,48)(37,54)(38,59)(39,63)(40,56)(41,64,47,77)(44,52,46,76)(45,73,70,96)(49,81)(50,84)(51,83)(53,58)(55,60)(57,80)(61,93)(62,82)(65,94,75,79)(66,87,78,86)(67,90,95,72)(69,71,92,74)(85,88)(89,91)Polycyclic presentation:
2^9|a^2=eh,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, 53283);