Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497426 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #22436 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,15,19,58)(4,21,24,10)(5,11)(6,30)(7,35)(8,13)(9,18,41,69)(12,33)(14,38)(16,60)(17,57,65,108)(20,50,36,59)(22,73,78,43)(23,80,83,94)(25,45,85,75)(26,88,89,99)(27,91)(28,54)(29,48)(31,53,70,42)(32,95)(34,55)(37,49)(39,62)(40,68,98,90)(44,84,74,114)(46,66,107,71)(47,92)(51,93)(52,67)(56,64)(61,111)(63,110)(72,117)(76,123)(77,104,126,122)(79,102,87,106)(81,97)(82,101,127,121)(86,103)(96,115)(100,112)(105,116)(109,113)(118,120)(119,124)(125,128);(1,3)(2,9)(4,16)(5,26)(6,31)(7,36)(8,19)(10,39)(11,46)(12,50)(13,41)(14,53)(15,55)(17,27)(18,67)(20,38)(21,72)(22,76)(23,81)(24,64)(25,86)(28,68)(29,89)(30,58)(32,66)(33,69)(34,70)(35,42)(37,90)(40,47)(43,100)(44,103)(45,105)(48,107)(49,57)(51,88)(52,59)(54,108)(56,80)(60,94)(61,77)(62,114)(63,85)(65,93)(71,92)(73,120)(74,110)(75,97)(78,125)(79,115)(82,112)(83,116)(84,117)(87,124)(91,99)(95,98)(96,101)(102,109)(104,128)(106,111)(113,126)(118,127)(119,121)(122,123);(1,4,8,24)(2,10,13,21)(3,16,19,64)(5,22,29,78)(6,23,34,83)(7,25,38,85)(9,39,41,72)(11,43,48,73)(12,44,52,74)(14,45,35,75)(15,56,58,60)(17,61,65,113)(18,62,69,117)(20,63,36,86)(26,76,89,125)(27,77,93,126)(28,79,37,87)(30,94,55,80)(31,81,70,116)(32,82,92,127)(33,84,67,114)(40,96,98,119)(42,97,53,105)(46,100,107,120)(47,101,95,121)(49,102,54,106)(50,103,59,110)(51,104,91,122)(57,109,108,111)(66,112,71,118)(68,115,90,124)(88,128,99,123);(1,5,8,29)(2,11,13,48)(3,17,19,65)(4,22,24,78)(6,32,34,92)(7,37,38,28)(9,40,41,98)(10,43,21,73)(12,51,52,91)(14,54,35,49)(15,57,58,108)(16,61,64,113)(18,68,69,90)(20,71,36,66)(23,82,83,127)(25,87,85,79)(26,70,89,31)(27,33,93,67)(30,95,55,47)(39,96,72,119)(42,99,53,88)(44,104,74,122)(45,106,75,102)(46,59,107,50)(56,109,60,111)(62,115,117,124)(63,118,86,112)(76,116,125,81)(77,84,126,114)(80,101,94,121)(97,123,105,128)(100,110,120,103)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,g^c=gi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404625);