Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C2×D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C2×C4⋊C32 |
| prime radical | C23 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,5,12,6,13,27,44,7,14,28,45,31,47,71,89,8,11,29,41,32,42,72,86,34,43,74,87,76,88,111,118,9,15,30,46,33,48,73,90,35,49,75,91,77,92,112,119,4,23,24,65,25,66,68,107,26,67,69,108,70,109,110,126)(3,16,19,51,20,52,59,96,21,53,60,97,62,98,103,123,22,37,61,80,63,81,104,114,64,82,105,115,106,116,125,127,17,50,54,93,55,94,99,120,56,95,100,121,101,122,124,128,18,10,36,38,57,39,78,83,58,40,79,84,102,85,113,117);(1,3,4,18,9,17,8,22)(2,10,11,16,15,37,23,50)(5,19,24,36,30,54,29,61)(6,20,25,57,33,55,32,63)(7,21,26,58,35,56,34,64)(12,38,41,51,46,80,65,93)(13,39,42,52,48,81,66,94)(14,40,43,53,49,82,67,95)(27,59,68,78,73,99,72,104)(28,60,69,79,75,100,74,105)(31,62,70,102,77,101,76,106)(44,83,86,96,90,114,107,120)(45,84,87,97,91,115,108,121)(47,85,88,98,92,116,109,122)(71,103,110,113,112,124,111,125)(89,117,118,123,119,127,126,128);(3,17)(10,37)(16,50)(18,22)(19,54)(20,55)(21,56)(36,61)(38,80)(39,81)(40,82)(51,93)(52,94)(53,95)(57,63)(58,64)(59,99)(60,100)(62,101)(78,104)(79,105)(83,114)(84,115)(85,116)(96,120)(97,121)(98,122)(102,106)(103,124)(113,125)(117,127)(123,128)Polycyclic presentation:
2^9|a^2=e,b^2=d,d^2=i,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,d^a=di,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72321739, #72321817, #72321834, … |
| maximal normal subgroup of | #72321739, #72321817, #72321834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60766);