Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506362 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28225 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,14)(3,15,20,55)(4,10,27,46)(5,28,31,84)(6,32,34,87)(7,35,33,13)(9,36,39,19)(11,47,49,29)(12,50,51,16)(17,60,63,52)(18,65,67,95)(21,70,69,56)(22,71,64,106)(23,75,79,124)(24,80,81,97)(25,45,26,44)(30,86,85,83)(37,62,92,88)(38,68,94,117)(40,57,54,90)(41,96,93,74)(42,99,102,76)(43,72,103,58)(48,59,82,104)(53,89,107,66)(61,112,114,98)(73,120,115,91)(77,123,78,122)(100,119,101,110)(105,113,111,125)(108,109,116,126)(118,127,128,121);(1,3,16,57,7,21,34,36,8,20,50,90,33,69,6,19)(2,9,32,56,13,40,51,55,14,39,87,70,35,54,12,15)(4,22,72,88,26,17,24,74,27,64,58,62,25,63,81,96)(5,18,59,109,30,68,47,89,31,67,104,126,85,117,29,66)(10,41,97,60,45,37,43,71,46,93,80,52,44,92,103,106)(11,38,83,108,48,95,84,107,49,94,86,116,82,65,28,53)(23,73,119,125,78,61,76,121,79,115,110,113,77,114,99,127)(42,98,122,111,101,91,75,118,102,112,123,105,100,120,124,128);(1,4)(2,10)(3,17)(5,23)(6,24)(7,26)(8,27)(9,37)(11,42)(12,43)(13,45)(14,46)(15,52)(16,58)(18,61)(19,62)(20,63)(21,64)(22,69)(25,33)(28,75)(29,76)(30,78)(31,79)(32,80)(34,81)(35,44)(36,88)(38,91)(39,92)(40,93)(41,54)(47,99)(48,101)(49,102)(50,72)(51,103)(53,105)(55,60)(56,106)(57,96)(59,110)(65,98)(66,113)(67,114)(68,115)(70,71)(73,117)(74,90)(77,85)(82,100)(83,123)(84,124)(86,122)(87,97)(89,125)(94,120)(95,112)(104,119)(107,111)(108,128)(109,127)(116,118)(121,126);(1,5,16,59,7,30,34,47,8,31,50,104,33,85,6,29)(2,11,32,83,13,48,51,84,14,49,87,86,35,82,12,28)(3,18,57,109,21,68,36,89,20,67,90,126,69,117,19,66)(4,23,58,110,26,78,81,99,27,79,72,119,25,77,24,76)(9,38,56,108,40,95,55,107,39,94,70,116,54,65,15,53)(10,42,80,123,45,101,103,124,46,102,97,122,44,100,43,75)(17,61,96,127,64,115,88,125,63,114,74,121,22,73,62,113)(37,91,106,128,93,112,60,111,92,120,71,118,41,98,52,105)Polycyclic presentation:
2^9|a^2=i,b^2=eg,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413559);