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 | #13528 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,5,13,36,61,44,69,9,17,6,14,37,62,81,73,10,18,40,65,43,68,101,91,7,15,38,63,8,16,39,64)(3,19,21,70,45,108,87,50,23,49,22,75,83,122,107,52,25,76,84,115,86,66,41,72,47,110,88,51,24,59,34,11)(4,26,28,56,46,109,42,12,32,54,99,71,105,67,35,57,33,95,100,120,102,60,103,58,30,93,29,92,31,94,98,55)(20,53,77,118,90,119,85,74,27,96,97,121,104,125,82,123,80,116,106,112,127,113,128,124,89,48,78,111,79,117,126,114);(1,3,10,25)(2,11,18,52)(4,27)(5,34,40,107)(6,41,38,87)(7,23,9,47)(8,45,37,86)(12,53)(13,59,65,122)(14,66,63,108)(15,50,17,72)(16,70,62,115)(19,73,76,64)(20,32)(21,81,84,39)(22,44,88,101)(24,43,83,36)(26,74)(28,85)(29,82)(30,80)(31,104)(33,89)(35,78)(42,77)(46,90)(48,57)(49,69,110,91)(51,68,75,61)(54,114)(55,96)(56,119)(58,116)(60,112)(67,111)(71,117)(79,105)(92,125)(93,123)(94,121)(95,124)(97,98)(99,126)(100,128)(102,127)(103,106)(109,118)(113,120);(1,4)(2,12)(3,20)(5,35)(6,42)(7,32)(8,46)(9,30)(10,33)(11,48)(13,60)(14,67)(15,57)(16,71)(17,55)(18,58)(19,74)(21,82)(22,85)(23,89)(24,90)(25,80)(26,91)(27,47)(28,81)(29,101)(31,36)(34,106)(37,102)(38,103)(39,100)(40,98)(41,78)(43,105)(44,99)(45,79)(49,114)(50,116)(51,117)(52,96)(53,72)(54,73)(56,61)(59,121)(62,92)(63,94)(64,93)(65,109)(66,112)(68,120)(69,95)(70,113)(75,125)(76,124)(77,107)(83,127)(84,126)(86,104)(87,97)(88,128)(108,118)(110,123)(111,122)(115,119)Polycyclic presentation:
2^9|a^2=d,b^2=degh,d^2=gh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egi,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,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, 60606);