Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13380 |
| prime radical | #33617 | C2 |
| 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 | |
258 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 258 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)(3,17)(4,23)(5,16)(6,32)(7,39)(8,12)(9,14)(10,19)(11,26)(13,36)(15,35)(18,63)(20,70)(21,73)(22,50)(24,81)(25,84)(27,80)(28,90)(29,45)(30,77)(31,57)(33,98)(34,101)(37,96)(38,97)(40,79)(41,52)(42,78)(43,107)(44,112)(46,75)(47,106)(48,103)(49,87)(51,86)(53,120)(54,105)(55,102)(56,94)(58,88)(59,85)(60,65)(61,72)(62,69)(64,108)(66,121)(67,76)(68,114)(71,127)(74,124)(82,110)(83,113)(89,119)(91,95)(92,118)(93,116)(99,109)(100,117)(104,123)(111,125)(115,128)(122,126);(1,3)(2,10)(4,24)(5,29)(6,33)(7,21)(8,19)(9,22)(11,43)(12,50)(13,53)(14,45)(15,46)(16,17)(18,27)(20,38)(23,76)(25,64)(26,65)(28,83)(30,68)(31,92)(32,95)(34,71)(35,69)(36,100)(37,91)(39,62)(40,104)(41,75)(42,67)(44,55)(47,110)(48,109)(49,111)(51,61)(52,118)(54,121)(56,117)(57,73)(58,115)(59,60)(63,78)(66,89)(70,96)(72,102)(74,105)(77,98)(79,113)(80,108)(81,84)(82,87)(85,125)(86,120)(88,99)(90,124)(93,128)(94,112)(97,114)(101,122)(103,126)(106,107)(116,127)(119,123);(1,4,5,25,9,27,8,42)(2,11,12,47,14,49,16,59)(3,18,19,64,22,24,29,67)(6,34,30,93,38,88,37,103)(7,40,31,89,41,105,35,28)(10,43,17,60,45,111,50,110)(13,54,51,119,55,79,56,90)(15,58,52,116,57,101,39,48)(20,71,68,126,33,99,91,128)(21,74,69,66,75,104,92,83)(23,77,78,32,80,96,84,97)(26,86,85,36,87,94,106,102)(44,113,61,123,53,121,117,124)(46,115,62,109,73,122,118,127)(63,114,76,70,81,95,108,98)(65,120,107,72,82,112,125,100)Polycyclic presentation:
2^9|c^2=d,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=ei,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, 60445);