Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #121 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12976 |
| prime radical | #575 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,6,14,7,15,33,55,8,16,34,56,36,57,83,104,9,12,35,38,37,48,84,85,5,13,29,51,30,52,79,101)(3,10,20,41,21,42,68,93,19,40,65,90,66,91,112,123,18,43,31,53,64,94,80,102,27,49,76,98,77,99,118,125)(4,22,25,70,26,71,75,116,24,47,73,54,74,97,117,103,23,50,32,81,72,100,82,119,28,11,44,45,78,46,95,96)(17,58,60,105,61,106,109,126,62,39,86,88,110,89,120,122,63,69,67,113,111,115,114,128,59,87,107,92,108,121,127,124);(1,3,6,31,7,21,33,80,8,19,34,76,36,66,83,118,9,18,35,20,37,64,84,68,5,27,29,65,30,77,79,112)(2,10,38,41,15,42,85,93,16,40,51,90,57,91,101,123,12,43,14,53,48,94,55,102,13,49,56,98,52,99,104,125)(4,17,32,60,26,61,82,109,24,62,44,86,74,110,95,120,23,63,25,67,72,111,75,114,28,59,73,107,78,108,117,127)(11,39,45,92,46,89,96,124,22,69,70,105,71,115,116,126,47,87,54,88,97,121,103,122,50,58,81,113,100,106,119,128);(1,4)(2,11)(3,17)(5,28)(6,32)(7,26)(8,24)(9,23)(10,39)(12,47)(13,50)(14,54)(15,46)(16,22)(18,63)(19,62)(20,67)(21,61)(25,35)(27,59)(29,73)(30,78)(31,60)(33,82)(34,44)(36,74)(37,72)(38,45)(40,69)(41,92)(42,89)(43,87)(48,97)(49,58)(51,70)(52,100)(53,88)(55,103)(56,81)(57,71)(64,111)(65,107)(66,110)(68,114)(75,84)(76,86)(77,108)(79,117)(80,109)(83,95)(85,96)(90,105)(91,115)(93,124)(94,121)(98,113)(99,106)(101,116)(102,122)(104,119)(112,127)(118,120)(123,126)(125,128)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59693);