Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30593 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
516 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)(4,21)(5,26)(6,31)(7,38)(8,13)(9,18)(10,23)(11,28)(12,36)(14,35)(16,59)(17,64)(19,68)(20,44)(22,46)(24,77)(25,85)(27,89)(29,92)(30,88)(32,93)(33,56)(34,57)(37,87)(39,53)(40,54)(41,82)(42,65)(43,96)(45,62)(47,97)(48,84)(49,90)(50,91)(51,98)(52,72)(55,69)(58,67)(60,99)(61,75)(63,102)(66,112)(70,100)(71,105)(73,111)(74,103)(76,79)(78,106)(80,109)(81,110)(83,107)(86,108)(94,104)(95,101)(113,121)(114,118)(115,127)(116,125)(117,126)(119,124)(120,122)(123,128);(1,3,7,20,8,18,35,67)(2,9,14,44,13,15,38,58)(4,16,25,63,24,61,39,73)(5,27,29,74,30,17,40,66)(6,32,36,96,37,19,69,72)(10,45,47,104,48,41,56,100)(11,49,51,105,50,42,57,101)(12,52,31,68,55,43,87,93)(21,75,53,102,77,59,85,111)(22,78,81,117,80,60,86,115)(23,82,33,94,84,62,97,70)(26,64,54,103,88,89,92,112)(28,65,34,71,91,90,98,95)(46,99,108,126,109,106,110,127)(76,113,119,125,120,118,107,128)(79,114,83,116,122,121,124,123);(1,4)(2,10)(3,16,18,61)(5,22)(6,33)(7,39)(8,24)(9,41,15,45)(11,46)(12,53)(13,48)(14,56)(17,60,27,78)(19,70,32,94)(20,73,67,63)(21,31)(23,36)(25,35)(26,76)(28,83)(29,86)(30,80)(34,79)(37,97)(38,47)(40,81)(42,99,49,106)(43,102,52,111)(44,104,58,100)(50,109)(51,110)(54,107)(55,85)(57,108)(59,93,75,68)(62,96,82,72)(64,118,89,113)(65,116,90,123)(66,117,74,115)(69,84)(71,114,95,121)(77,87)(88,120)(91,124)(92,119)(98,122)(101,127,105,126)(103,125,112,128);(1,5,8,30)(2,11,13,50)(3,17,18,27)(4,22,24,80)(6,34,37,98)(7,40,35,29)(9,42,15,49)(10,46,48,109)(12,54,55,92)(14,57,38,51)(16,60,61,78)(19,71,32,95)(20,74,67,66)(21,76,77,120)(23,83,84,124)(25,86,39,81)(26,87,88,31)(28,69,91,36)(33,79,97,122)(41,99,45,106)(43,103,52,112)(44,105,58,101)(47,110,56,108)(53,107,85,119)(59,113,75,118)(62,116,82,123)(63,117,73,115)(64,68,89,93)(65,96,90,72)(70,114,94,121)(100,127,104,126)(102,125,111,128)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416541);