Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2184 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12731 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 614 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)(3,16)(4,23,26,50)(5,29)(6,12)(7,36)(8,15)(9,18)(10,25,44,28)(11,33)(13,31)(14,35)(17,58,61,112)(19,56)(20,68)(21,70)(22,57)(24,74,78,122)(27,83,82,45)(30,52)(32,86)(34,88)(37,46)(38,51)(39,96,97,116)(40,63)(41,90)(42,65)(43,101,103,125)(47,92)(48,107)(49,95)(53,93)(54,64)(55,60,111,66)(59,71,114,99)(62,102,115,128)(67,105)(69,108)(72,120,106,79)(73,81,109,76)(75,98,124,127)(77,121,80,85)(84,113,117,118)(87,94)(89,100)(91,110)(104,123,119,126);(1,3)(2,9)(4,17)(5,30)(6,19)(7,37)(8,22)(10,39)(11,46)(12,40)(13,52)(14,54)(15,42)(16,29)(18,33)(20,31)(21,35)(23,71)(24,75)(25,60)(26,61)(27,84)(28,66)(32,89)(34,91)(36,41)(38,67)(43,102)(44,97)(45,104)(47,68)(48,108)(49,105)(50,99)(51,100)(53,110)(55,85)(56,86)(57,88)(58,74)(59,79)(62,76)(63,92)(64,107)(65,95)(69,93)(70,87)(72,113)(73,98)(77,123)(78,124)(80,126)(81,115)(82,117)(83,119)(90,94)(96,101)(103,128)(106,118)(109,127)(111,121)(112,122)(114,120)(116,125);(1,4)(2,10)(3,17)(5,24)(6,28)(7,27)(8,26)(9,39)(11,43)(12,50)(13,45)(14,23)(15,44)(16,55)(18,59)(19,66)(20,62)(21,60)(22,61)(25,35)(29,85)(30,75)(31,76)(32,80)(33,79)(34,78)(36,73)(37,84)(38,82)(40,99)(41,98)(42,97)(46,102)(47,106)(48,72)(49,103)(51,109)(52,104)(53,83)(54,71)(56,112)(57,111)(58,70)(63,116)(64,96)(65,114)(67,117)(68,118)(69,115)(74,87)(77,94)(81,93)(86,122)(88,121)(89,126)(90,123)(91,124)(92,125)(95,120)(100,127)(101,107)(105,128)(108,113)(110,119)Polycyclic presentation:
2^9|a^2=f,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi,e^c=ei,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59369);