Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152169 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12731 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 104 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)(4,21,23,10)(5,26)(6,32)(7,36)(8,12)(9,17)(11,29)(13,28)(14,34)(16,55,57,52)(18,63)(19,66)(20,53)(22,71,74,82)(24,79,77,92)(25,44,78,70)(27,49)(30,90)(31,51)(33,45)(35,93)(37,54)(38,56,67,111)(39,62)(40,86)(41,85)(42,75,68,110)(43,73,69,115)(46,60)(47,103)(48,64)(50,61)(58,114,113,116)(59,106,81,109)(65,104)(72,107,120,100)(76,125,124,119)(80,108,127,98)(83,88)(84,89)(87,105)(91,97)(94,102)(95,122,117,126)(96,121,101,112)(99,128,118,123);(1,3,8,20)(2,9,12,39)(4,16,23,57)(5,27,31,87)(6,33,35,94)(7,37,34,19)(10,38,21,67)(11,45,46,102)(13,49,48,105)(14,50,36,41)(15,51,53,26)(17,60,62,29)(18,64,65,28)(22,72,74,120)(24,80,77,127)(25,81,78,59)(30,91,89,86)(32,40,93,97)(42,98,68,108)(43,100,69,107)(44,101,70,96)(47,104,83,63)(52,71,55,82)(54,84,66,90)(56,110,111,75)(58,115,113,73)(61,88,85,103)(76,126,124,122)(79,117,92,95)(99,114,118,116)(106,125,109,119)(112,123,121,128);(1,4)(2,10)(3,16)(5,22)(6,24)(7,25)(8,23)(9,38)(11,42)(12,21)(13,43)(14,44)(15,52)(17,56)(18,58)(19,59)(20,57)(26,82)(27,72)(28,73)(29,75)(30,76)(31,74)(32,92)(33,80)(34,78)(35,77)(36,70)(37,81)(39,67)(40,95)(41,96)(45,98)(46,68)(47,99)(48,69)(49,100)(50,101)(51,71)(53,55)(54,106)(60,110)(61,112)(62,111)(63,116)(64,115)(65,113)(66,109)(79,93)(83,118)(84,125)(85,121)(86,122)(87,120)(88,123)(89,124)(90,119)(91,126)(94,127)(97,117)(102,108)(103,128)(104,114)(105,107)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59366);