Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1339 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2151 | C22 |
| socle | C2 | #7080 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(1,2,5,12,8,15,37,63,9,17,38,65,43,16,69,64)(3,18,20,71,23,73,83,125,25,74,84,126,24,89,34,102)(4,26,29,90,32,55,97,119,33,66,98,118,100,91,114,123)(6,39,42,105,44,68,103,61,28,14,35,62,7,13,36,67)(10,46,47,79,50,80,110,78,51,107,111,19,70,75,59,77)(11,52,54,101,56,115,120,30,58,99,60,31,57,96,121,41)(21,85,87,128,22,88,82,122,45,53,81,116,40,72,106,117)(27,92,93,124,76,48,95,112,94,49,113,109,86,104,127,108);(1,3)(2,10)(4,27)(5,34)(6,40)(7,45)(8,23)(9,25)(11,53)(12,59)(13,49)(14,48)(15,50)(16,70)(17,51)(18,55)(19,31)(20,37)(21,44)(22,28)(24,43)(26,89)(29,95)(30,78)(32,76)(33,94)(35,87)(36,82)(38,83)(39,104)(41,77)(42,81)(46,96)(47,63)(52,80)(54,117)(56,72)(57,88)(58,85)(60,122)(61,108)(62,124)(64,111)(65,110)(66,73)(67,112)(68,92)(69,84)(71,90)(74,91)(75,99)(79,101)(86,100)(93,114)(97,113)(98,127)(102,123)(103,106)(105,109)(107,115)(116,121)(118,126)(119,125)(120,128)Polycyclic presentation:
2^9|a^2=d,c^2=fhi,d^2=g,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1006);