Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13147 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,7,15,10,17,9,46)(3,18,22,65,25,67,24,86)(4,26,29,89,32,60,31,99)(5,33,35,48,38,102,37,14)(6,39,42,59,43,101,36,13)(8,45,44,61,41,100,34,16)(11,47,51,78,53,77,64,20)(12,54,56,40,58,96,103,28)(19,68,71,106,74,115,73,49)(21,79,82,114,83,52,76,63)(23,85,84,116,81,50,75,66)(27,90,92,88,94,126,93,57)(30,98,97,109,95,127,91,55)(62,108,111,80,113,122,125,70)(69,87,104,117,120,107,119,112)(72,124,123,128,121,105,118,110);(1,3,6,21,41,81,5,20,7,22,42,82,34,75,35,47,10,25,43,83,8,23,38,78,9,24,36,76,44,84,37,77)(2,11,48,85,61,79,59,67,46,64,33,66,45,63,39,65,17,53,14,50,16,52,13,18,15,51,102,116,100,114,101,86)(4,19,40,80,95,121,27,69,29,71,96,122,91,118,92,104,32,74,28,70,30,72,94,120,31,73,54,108,97,123,93,119)(12,49,26,87,88,124,109,113,103,115,99,112,90,110,98,111,58,106,60,107,57,105,55,62,56,68,89,117,126,128,127,125);(1,4,5,27,34,91,43,28,9,31,37,93,41,95,42,96,10,32,38,94,44,97,6,40,7,29,35,92,8,30,36,54)(2,12,13,55,45,90,48,26,15,56,39,98,61,88,102,89,17,58,59,109,100,126,14,60,46,103,101,127,16,57,33,99)(3,19,20,69,75,118,83,70,24,73,77,119,81,121,82,122,25,74,78,120,84,123,21,80,22,71,47,104,23,72,76,108)(11,49,18,62,63,110,85,87,51,68,65,111,79,124,116,117,53,106,67,113,114,128,50,107,64,115,86,125,52,105,66,112)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gh,h^a=hi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320849, #72320873 |
| maximal normal subgroup of | #72320566, #72320849, #72320873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60034);