Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13165 |
| prime radical | C23 | C2×Q32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
170 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 128 points:
(1,2,5,14,9,16,8,12)(3,17,21,42,23,48,19,10)(4,24,26,76,30,78,29,74)(6,33,31,88,36,91,35,90)(7,37,32,54,38,50,34,15)(11,43,45,85,47,84,44,27)(13,51,52,81,53,80,49,25)(18,58,62,104,64,107,60,73)(20,67,68,117,69,118,65,89)(22,71,70,101,72,41,66,57)(28,86,82,121,87,120,83,77)(39,61,55,95,93,113,97,111)(40,59,56,102,94,110,100,109)(46,79,75,96,99,123,98,122)(63,115,114,127,116,119,112,106)(92,124,103,108,105,125,126,128);(1,3,5,21,9,23,8,19)(2,10,14,17,16,42,12,48)(4,18,26,62,30,64,29,60)(6,20,31,68,36,69,35,65)(7,22,32,70,38,72,34,66)(11,39,45,55,47,93,44,97)(13,40,52,56,53,94,49,100)(15,41,37,57,54,71,50,101)(24,73,76,58,78,104,74,107)(25,59,51,102,81,110,80,109)(27,61,43,95,85,113,84,111)(28,63,82,114,87,116,83,112)(33,89,88,67,91,117,90,118)(46,92,75,103,99,105,98,126)(77,119,86,106,121,115,120,127)(79,108,96,125,123,128,122,124);(1,4,25,79,7,28,31,43,5,26,51,96,32,82,36,85,9,30,81,123,38,87,35,84,8,29,80,122,34,83,6,27)(2,11,33,77,15,46,49,74,12,44,90,120,50,98,53,78,16,47,91,121,54,99,52,76,14,45,88,86,37,75,13,24)(3,18,59,108,22,63,68,95,21,62,102,125,70,114,69,113,23,64,110,128,72,116,65,111,19,60,109,124,66,112,20,61)(10,39,89,119,41,92,100,107,48,97,118,127,101,126,94,104,42,93,117,115,71,105,56,58,17,55,67,106,57,103,40,73)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320842, #72320880, … |
| maximal normal subgroup of | #72320566, #72320842, #72320880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60055);