Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13494 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,10,40,37,55,96,119,8,13,35,59,38,61,86,118,9,16,31,90,32,57,34,58,5,29,6,14,7,15,43,78)(3,17,41,46,70,47,104,127,19,11,44,99,75,111,126,124,22,48,91,109,72,112,73,122,30,49,20,63,21,64,102,108)(4,23,76,53,36,97,94,12,25,82,88,60,89,117,42,54,27,81,95,115,87,100,33,52,39,79,26,80,28,56,92,51)(18,65,123,110,74,116,93,62,69,77,84,114,85,128,101,50,24,83,125,120,98,107,71,113,66,105,67,106,68,121,103,45);(1,3)(2,11)(4,24)(5,30)(6,20)(7,21)(8,19)(9,22)(10,41)(12,50)(13,48)(14,46)(15,47)(16,49)(17,29)(18,27)(23,77)(25,66)(26,84)(28,85)(31,91)(32,72)(33,93)(34,73)(35,44)(36,98)(37,70)(38,75)(39,69)(40,99)(42,103)(43,102)(45,52)(51,62)(53,114)(54,113)(55,111)(56,116)(57,64)(58,108)(59,109)(60,120)(61,112)(63,90)(65,79)(67,88)(68,89)(71,94)(74,87)(76,125)(78,127)(80,110)(81,105)(82,83)(86,126)(92,101)(95,123)(96,104)(97,128)(100,121)(106,115)(107,117)(118,122)(119,124);(1,4,5,25,9,27,8,39)(2,12,13,51,16,52,29,54)(3,18,19,66,22,24,30,69)(6,33,31,92,35,94,10,42)(7,36,32,89,38,87,37,28)(11,45,17,62,49,50,48,113)(14,56,40,100,59,117,90,97)(15,60,55,53,61,80,57,115)(20,71,41,101,44,93,91,103)(21,74,70,68,75,98,72,85)(23,78,79,119,81,118,82,58)(26,86,76,96,88,43,95,34)(46,107,63,121,109,116,99,128)(47,110,64,114,112,120,111,106)(65,124,77,127,83,108,105,122)(67,102,125,104,84,126,123,73)Polycyclic presentation:
2^9|a^2=eh,c^2=d,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60573);