Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13482 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,17,22,10)(4,23,28,62)(5,13)(6,14)(7,15)(8,16)(9,12)(11,25,47,41)(18,66)(19,61,40,43)(20,64,73,44)(21,65,75,45)(24,78)(26,77,83,117)(27,60,85,113)(29,52)(30,53)(31,89,91,54)(32,92,94,120)(33,56)(34,57)(35,58)(36,50)(37,99,100,59)(38,101,102,49)(39,51)(42,67)(46,79)(48,82,109,93)(55,87,112,98)(63,69)(68,107)(70,103)(71,116,90,105)(72,111)(74,121)(76,81)(80,108)(84,122)(86,119,97,114)(88,115)(95,118)(96,110)(104,124)(106,126)(123,125)(127,128);(1,3)(2,10)(4,24)(5,19)(6,31)(7,37)(8,40)(9,22)(11,46)(12,17)(13,43)(14,54)(15,59)(16,61)(18,26)(20,34)(21,33)(23,76)(25,79)(27,84)(28,81)(29,86)(30,75)(32,74)(35,97)(36,91)(38,72)(39,100)(41,80)(42,48)(44,57)(45,56)(47,108)(49,111)(50,89)(51,99)(52,114)(53,65)(55,107)(58,119)(60,106)(62,78)(63,109)(64,110)(66,117)(67,82)(68,87)(69,93)(70,83)(71,88)(73,96)(77,103)(85,126)(90,95)(92,104)(94,124)(98,125)(101,128)(102,127)(105,115)(112,123)(113,122)(116,118)(120,121);(1,4)(2,11,12,47)(3,18)(5,25)(6,32)(7,38)(8,41)(9,28)(10,42,17,63)(13,23,16,62)(14,55,50,112)(15,60,51,113)(19,67)(20,72)(21,74)(22,70)(24,31)(26,34)(27,33)(29,87)(30,85)(35,98)(36,94)(37,68)(39,102)(40,69)(43,103,61,66)(44,106,64,122)(45,107,65,123)(46,89,108,54)(48,110,109,57)(49,53,101,56)(52,92,58,120)(59,104,99,121)(71,126)(73,127)(75,124)(76,119,78,114)(77,118,117,115)(79,86)(80,97)(81,91)(82,88)(83,96)(84,90)(93,95)(100,125)(105,128,116,111)Polycyclic presentation:
2^9|a^2=eh,c^2=d,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fg,g^b=gh,h^b=hi,f^c=fg,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, 60559);