Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| 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×D4 | C2×C16 |
| Frattini, cosocle | #2113 | C22 |
| socle | C2 | #7054 |
| prime radical | #585 | C8 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,12,7,14,30,52,8,13,29,54,37,55,83,107,9,15,31,53,38,56,84,106,6,32,35,51,36,85,88,105)(3,16,18,57,20,59,67,109,19,58,27,79,69,110,80,121,21,60,68,92,70,111,116,124,33,71,66,108,86,117,115,128)(4,22,23,72,25,73,76,118,24,11,45,46,77,47,99,100,26,49,28,81,78,103,82,122,34,48,75,101,87,102,120,127)(10,39,40,89,42,91,94,64,43,90,93,114,97,123,125,62,44,74,95,113,98,119,126,17,41,61,50,63,96,112,104,65);(1,3)(2,10)(4,17)(5,27)(6,33)(7,20)(8,19)(9,21)(11,16)(12,50)(13,43)(14,42)(15,44)(18,35)(22,71)(23,74)(24,65)(25,63)(26,64)(28,39)(29,68)(30,80)(31,66)(32,41)(34,62)(36,86)(37,69)(38,70)(40,54)(45,61)(46,79)(47,59)(48,60)(49,58)(51,95)(52,104)(53,93)(55,97)(56,98)(57,72)(67,88)(73,117)(75,90)(76,119)(77,89)(78,114)(81,92)(82,91)(83,116)(84,115)(85,96)(87,113)(94,107)(99,112)(100,121)(101,108)(102,111)(103,110)(105,126)(106,125)(109,118)(120,123)(122,124)(127,128)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=dei,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, 875);