Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153111 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13325 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,15)(4,21)(5,27)(6,34)(7,39)(8,12)(9,17)(10,23)(11,31)(13,30)(14,37)(16,42)(18,73)(19,77)(20,62)(22,48)(24,87)(25,89)(26,47)(28,56)(29,57)(32,98)(33,60)(35,51)(36,52)(38,101)(40,63)(41,81)(43,72)(44,93)(45,92)(46,68)(49,95)(50,85)(53,69)(54,115)(55,74)(58,71)(59,86)(61,65)(64,79)(66,108)(67,109)(70,104)(75,105)(76,116)(78,106)(80,84)(82,103)(83,111)(88,117)(90,96)(91,97)(94,118)(99,107)(100,112)(102,114)(110,122)(113,125)(119,126)(120,123)(121,124)(127,128);(1,3,8,20)(2,9,12,43)(4,16,26,68)(5,28,33,94)(6,35,38,102)(7,40,37,19)(10,46,48,42)(11,51,53,114)(13,56,55,118)(14,58,39,45)(15,60,62,27)(17,69,72,31)(18,74,76,30)(21,79,47,61)(22,82,23,70)(24,75,88,122)(25,78,41,67)(29,65,84,64)(32,99,97,93)(34,44,101,107)(36,66,83,125)(49,113,112,108)(50,106,59,109)(52,103,111,104)(54,116,90,73)(57,110,80,105)(63,91,77,98)(71,96,92,115)(81,120,89,121)(85,126,86,127)(87,128,117,119)(95,123,100,124);(1,4)(2,10)(3,16)(5,29)(6,36)(7,41)(8,26)(9,42)(11,52)(12,48)(13,57)(14,59)(15,61)(17,70)(18,75)(19,78)(20,68)(21,27)(22,69)(23,31)(24,30)(25,37)(28,65)(32,100)(33,84)(34,49)(35,66)(38,83)(39,50)(40,67)(43,46)(44,108)(45,109)(47,60)(51,104)(53,111)(54,117)(55,80)(56,105)(58,106)(62,79)(63,121)(64,94)(71,127)(72,82)(73,119)(74,88)(76,122)(77,120)(81,98)(85,96)(86,115)(87,90)(89,91)(92,126)(93,123)(95,97)(99,124)(101,112)(102,125)(103,114)(107,113)(110,118)(116,128)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dgi,e^c=eg,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60308);