Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C23 | C2×D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 136 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,17,24,43)(12,28,33,73)(13,34,37,85)(14,38,40,87)(15,41,39,21)(18,47,50,30)(19,51,53,29)(20,54,55,35)(23,56,58,104)(25,65,62,117)(26,68,63,118)(27,69,64,46)(31,79,81,126)(32,82,80,72)(36,86,75,84)(42,89,90,59)(44,95,93,57)(45,97,94,66)(48,101,102,125)(49,78,88,100)(52,76,83,103)(60,115,111,136)(61,116,112,107)(67,121,109,120)(70,99,123,74)(71,98,124,77)(91,131,130,135)(92,114,122,129)(96,110,119,132)(105,128,133,108)(106,127,134,113);(2,5)(6,8)(9,11,16,24)(10,17,22,43)(12,23,33,58)(13,25,37,62)(14,26,40,63)(15,27,39,64)(18,42,50,90)(19,44,53,93)(20,45,55,94)(21,46,41,69)(28,56,73,104)(29,57,51,95)(30,59,47,89)(31,60,81,111)(32,61,80,112)(34,65,85,117)(35,66,54,97)(36,67,75,109)(38,68,87,118)(48,91,102,130)(49,92,88,122)(52,96,83,119)(70,105,123,133)(71,106,124,134)(72,107,82,116)(74,108,99,128)(76,110,103,132)(77,113,98,127)(78,114,100,129)(79,115,126,136)(84,120,86,121)(101,131,125,135);(9,12,29,74,14,31,75,100,39,80,103,101,54,98,37,47,16,33,51,99,40,81,36,78,15,32,76,125,35,77,13,30)(10,18,34,71,20,48,83,82,41,88,86,126,87,123,53,73,22,50,85,124,55,102,52,72,21,49,84,79,38,70,19,28)(11,23,57,108,26,60,109,129,64,112,132,131,97,127,62,89,24,58,95,128,63,111,67,114,27,61,110,135,66,113,25,59)(17,42,65,106,45,91,119,116,69,122,121,136,118,133,93,104,43,90,117,134,94,130,96,107,46,92,120,115,68,105,44,56);(1,3,4,7)(2,6,5,8)(11,24)(17,43)(23,58)(25,62)(26,63)(27,64)(42,90)(44,93)(45,94)(46,69)(56,104)(57,95)(59,89)(60,111)(61,112)(65,117)(66,97)(67,109)(68,118)(91,130)(92,122)(96,119)(105,133)(106,134)(107,116)(108,128)(110,132)(113,127)(114,129)(115,136)(120,121)(131,135)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420271);