Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C16 |
| Frattini, cosocle | #2113 | C22 |
| socle | C2 | #7054 |
| prime radical | #585 | C8 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,7,14,31,54,8,13,30,56,38,57,85,110,9,15,32,55,39,58,86,109,6,33,36,53,37,87,90,108)(3,16,18,59,20,61,66,111,19,60,68,102,69,112,116,127,21,47,67,93,70,101,115,124,34,88,28,79,80,120,81,121)(4,22,24,71,26,72,76,117,25,11,46,48,77,49,100,103,27,51,29,82,78,106,84,122,35,50,75,104,89,105,119,128)(10,40,41,91,43,92,94,17,44,62,52,64,98,113,107,65,45,83,95,114,99,123,125,23,42,73,96,74,97,118,126,63);(1,3)(2,10)(4,23)(5,28)(6,34)(7,20)(8,19)(9,21)(11,47)(12,52)(13,44)(14,43)(15,45)(16,50)(17,27)(18,30)(22,60)(24,40)(25,63)(26,74)(29,83)(31,81)(32,68)(33,42)(35,65)(36,67)(37,80)(38,69)(39,70)(41,53)(46,62)(48,102)(49,101)(51,88)(54,107)(55,96)(56,95)(57,98)(58,99)(59,71)(61,105)(64,78)(66,85)(72,112)(73,75)(76,92)(77,91)(79,104)(82,93)(84,123)(86,116)(87,97)(89,114)(90,115)(94,108)(100,113)(103,127)(106,120)(109,126)(110,125)(111,117)(118,119)(121,128)(122,124)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=de,e^b=ei,g^b=gi,d^c=dgi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 876);