Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153097 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13304 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,8,14)(3,15,21,63)(4,22,28,50)(5,29,35,91)(6,36,37,12)(7,38,40,102)(9,41,46,17)(10,27,51,25)(11,52,56,34)(13,57,59,32)(16,64,68,124)(18,62,19,61)(20,73,74,119)(23,77,66,60)(24,80,31,96)(26,49,87,116)(30,58,95,109)(33,100,101,89)(39,53,72,120)(42,105,108,78)(43,70,44,71)(45,93,110,94)(47,69,106,65)(48,112,54,121)(55,99,90,117)(67,126,127,118)(75,98,115,84)(76,86,103,82)(79,122,104,113)(81,123,97,111)(83,114,85,88)(92,128,125,107);(1,3)(2,9)(4,23)(5,30)(6,18)(7,39)(8,21)(10,47)(11,53)(12,43)(13,58)(14,46)(15,29)(16,25)(17,34)(19,37)(20,32)(22,42)(24,81)(26,67)(27,68)(28,66)(31,97)(33,93)(35,95)(36,44)(38,45)(40,72)(41,52)(48,113)(49,107)(50,108)(51,106)(54,122)(55,119)(56,120)(57,74)(59,109)(60,96)(61,89)(62,100)(63,91)(64,114)(65,121)(69,112)(70,99)(71,117)(73,90)(75,126)(76,123)(77,80)(78,84)(79,82)(83,125)(85,92)(86,104)(87,127)(88,124)(94,101)(98,105)(102,110)(103,111)(115,118)(116,128);(1,4)(2,10)(3,16)(5,31)(6,27)(7,26)(8,28)(9,42)(11,54)(12,50)(13,49)(14,51)(15,60)(17,69)(18,23)(19,66)(20,67)(21,68)(22,36)(24,35)(25,37)(29,88)(30,92)(32,82)(33,83)(34,98)(38,103)(39,104)(40,87)(41,65)(43,47)(44,106)(45,107)(46,108)(48,56)(52,84)(53,118)(55,75)(57,86)(58,123)(59,116)(61,124)(62,64)(63,77)(70,105)(71,78)(72,79)(73,113)(74,127)(76,102)(80,89)(81,94)(85,101)(90,115)(91,114)(93,97)(95,125)(96,100)(99,121)(109,111)(110,128)(112,117)(119,122)(120,126)Polycyclic presentation:
2^9|a^2=i,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60294);