Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503268 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25779 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 176 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 — 4 generators on 128 points:
(1,2,8,13)(3,15,20,52)(4,10,25,43)(5,27,30,80)(6,31,33,84)(7,34,32,14)(9,35,38,19)(11,44,45,28)(12,47,48,16)(17,57,22,49)(18,62,64,91)(21,67,66,53)(23,71,73,119)(24,75,77,122)(26,78,76,42)(29,83,82,81)(36,60,88,69)(37,65,90,114)(39,54,51,86)(40,92,94,72)(41,58,95,55)(46,56,79,96)(50,85,100,63)(59,109,68,112)(61,99,70,107)(74,121,120,118)(87,116,124,111)(89,104,98,102)(93,105,117,108)(97,110,106,115)(101,103,113,123)(125,127,128,126);(1,3,16,54,7,21,33,35,8,20,47,86,32,66,6,19)(2,9,31,53,14,39,48,52,13,38,84,67,34,51,12,15)(4,22,55,104,26,70,77,60,25,17,58,102,76,61,24,69)(5,18,56,103,29,65,44,85,30,64,96,123,82,114,28,63)(10,36,75,99,42,89,95,57,43,88,122,107,78,98,41,49)(11,37,81,101,46,91,80,100,45,90,83,113,79,62,27,50)(23,68,105,127,74,116,92,110,73,59,108,126,120,111,72,115)(40,87,118,125,93,109,119,106,94,124,121,128,117,112,71,97);(1,4)(2,10)(3,17,20,22)(5,23)(6,24)(7,26)(8,25)(9,36,38,88)(11,40)(12,41)(13,43)(14,42)(15,49,52,57)(16,55)(18,59,64,68)(19,60,35,69)(21,61,66,70)(27,71)(28,72)(29,74)(30,73)(31,75)(32,76)(33,77)(34,78)(37,87,90,124)(39,89,51,98)(44,92)(45,94)(46,93)(47,58)(48,95)(50,97,100,106)(53,99,67,107)(54,102,86,104)(56,105)(62,112,91,109)(63,110,85,115)(65,111,114,116)(79,117)(80,119)(81,118)(82,120)(83,121)(84,122)(96,108)(101,125,113,128)(103,126,123,127);(1,5,16,56,7,29,33,44,8,30,47,96,32,82,6,28)(2,11,31,81,14,46,48,80,13,45,84,83,34,79,12,27)(3,18,54,103,21,65,35,85,20,64,86,123,66,114,19,63)(4,23,55,105,26,74,77,92,25,73,58,108,76,120,24,72)(9,37,53,101,39,91,52,100,38,90,67,113,51,62,15,50)(10,40,75,118,42,93,95,119,43,94,122,121,78,117,41,71)(17,59,102,126,61,111,69,115,22,68,104,127,70,116,60,110)(36,87,99,125,89,109,57,106,88,124,107,128,98,112,49,97)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=f,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410465);