Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| 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 | |
187 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,5,10,6,11,7,21)(3,12,14,24,16,28,15,8)(4,17,19,27,20,9,22,26)(13,29,18,32,30,25,31,23)(33,34,37,45,41,49,38,46,42,50,67,82,40,48,39,47)(35,51,53,73,56,71,54,74,57,83,68,72,55,81,66,43)(36,58,60,44,64,76,62,79,65,77,69,80,63,84,61,78)(52,86,87,85,89,95,91,96,90,94,93,75,88,70,59,92);(1,3,7,14,6,16,5,15)(2,8,21,12,11,24,10,28)(4,18,22,30,20,31,19,13)(9,25,27,23,17,29,26,32)(33,35)(34,43)(36,59,65,91)(37,66)(38,68)(39,53)(40,56)(41,55)(42,57)(44,75,80,85)(45,81)(46,83)(47,51)(48,73)(49,72)(50,74)(52,62,90,61)(54,67)(58,70,77,95)(60,88,69,89)(63,87,64,93)(71,82)(76,94,84,86)(78,92,79,96);(1,4)(2,9)(3,13)(5,19)(6,20)(7,22)(8,23)(10,26)(11,17)(12,29)(14,18)(15,31)(16,30)(21,27)(24,32)(25,28)(33,36,42,65)(34,44,50,80)(35,52,57,90)(37,62,67,61)(38,60,39,69)(40,64,41,63)(43,70,74,95)(45,77,82,58)(46,76,47,84)(48,79,49,78)(51,85,83,75)(53,91,68,59)(54,87,66,93)(55,89,56,88)(71,92,81,96)(72,86,73,94)Polycyclic presentation:
2^9|a^2=d,b^2=dei,c^2=i,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53399);