Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22×D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C22×C22⋊C16 |
| prime radical | #585 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 352 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 352 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 — 4 generators on 128 points:
(1,2,6,13,7,14,32,51,8,15,33,52,35,54,80,101,9,16,34,53,36,55,81,102,5,12,29,48,30,49,77,99)(3,17,20,57,21,58,65,105,19,41,63,50,64,88,110,100,22,47,31,78,66,98,79,117,28,10,37,39,76,40,83,87)(4,11,25,44,26,45,73,95,27,46,74,96,75,97,116,125,23,42,67,89,68,90,111,121,24,43,70,92,71,93,113,123)(18,56,61,103,62,104,109,126,60,84,107,94,108,119,127,124,59,91,72,114,106,122,115,128,69,38,82,85,112,86,118,120);(1,3)(2,10)(4,18)(5,28)(6,31)(7,21)(8,19)(9,22)(11,38)(12,47)(13,50)(14,40)(15,17)(16,41)(20,34)(23,59)(24,69)(25,72)(26,62)(27,60)(29,63)(30,76)(32,79)(33,37)(35,64)(36,66)(39,53)(42,84)(43,91)(44,94)(45,86)(46,56)(48,57)(49,98)(51,100)(52,78)(54,58)(55,88)(61,67)(65,81)(68,106)(70,107)(71,112)(73,115)(74,82)(75,108)(77,110)(80,83)(85,89)(87,102)(90,119)(92,103)(93,122)(95,124)(96,114)(97,104)(99,105)(101,117)(109,111)(113,127)(116,118)(120,121)(123,126)(125,128);(1,4)(2,11)(3,18)(5,24)(6,25)(7,26)(8,27)(9,23)(10,38)(12,43)(13,44)(14,45)(15,46)(16,42)(17,56)(19,60)(20,61)(21,62)(22,59)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,67)(35,75)(36,68)(37,82)(39,85)(40,86)(41,84)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,89)(54,97)(55,90)(57,103)(58,104)(63,107)(64,108)(65,109)(66,106)(76,112)(77,113)(78,114)(79,115)(80,116)(81,111)(83,118)(87,120)(88,119)(98,122)(99,123)(100,124)(101,125)(102,121)(105,126)(110,127)(117,128);(4,23)(11,42)(18,59)(24,27)(25,67)(26,68)(38,84)(43,46)(44,89)(45,90)(56,91)(60,69)(61,72)(62,106)(70,74)(71,75)(73,111)(82,107)(85,94)(86,119)(92,96)(93,97)(95,121)(103,114)(104,122)(108,112)(109,115)(113,116)(118,127)(120,124)(123,125)(126,128)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=be,e^b=ei,f^b=fi,d^c=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419753);