Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,12,7,10,5,11)(3,13,15,26,16,8,21,23)(4,17,19,9,20,25,18,24)(14,28,29,22,30,32,31,27)(33,34,39,45,38,44,62,43,40,46,65,77,61,76,37,59)(35,47,51,69,50,74,89,78,52,79,91,41,63,68,49,67)(36,53,56,75,58,42,60,71,57,73,64,70,55,72,54,80)(48,82,85,94,87,81,88,66,86,95,90,92,84,93,83,96);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,22)(10,13)(11,23)(12,26)(17,32)(18,29)(19,31)(20,30)(24,27)(25,28)(33,35,40,52)(34,41,46,69)(36,48,57,86)(37,51,62,91)(38,63,61,50)(39,49,65,89)(42,66,72,96)(43,74,59,68)(44,78,76,67)(45,79,77,47)(53,92,73,94)(54,85,60,90)(55,87,58,84)(56,83,64,88)(70,82,75,95)(71,81,80,93);(1,4,7,20)(2,9,10,24)(3,14,16,30)(5,19,6,18)(8,22,13,27)(11,25,12,17)(15,31,21,29)(23,28,26,32)(33,36,40,57)(34,42,46,72)(35,48,52,86)(37,60,62,54)(38,58,61,55)(39,64,65,56)(41,66,69,96)(43,75,59,70)(44,73,76,53)(45,80,77,71)(47,81,79,93)(49,88,89,83)(50,87,63,84)(51,90,91,85)(67,92,78,94)(68,82,74,95)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=eg,f^c=fg53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38970);