Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472908 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16841 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,8,16,33)(4,17,21,39)(5,22,24,10)(6,26,29,11)(9,27,36,20)(13,41,18,51)(14,31,45,55)(15,47,50,32)(19,52,23,56)(25,38,28,40)(30,48,34,43)(35,53,37,54)(42,62,44,61)(46,60,49,59)(57,64,58,63);(1,3)(2,8)(4,18,21,13)(5,14)(6,15)(7,16)(9,34,36,30)(10,31)(11,32)(12,33)(17,51,39,41)(19,44,23,42)(20,48,27,43)(22,55)(24,45)(25,46)(26,47)(28,49)(29,50)(35,58,37,57)(38,59)(40,60)(52,62,56,61)(53,64,54,63);(1,4)(2,9)(3,13)(5,23)(6,27)(7,21)(8,30)(10,37)(11,39)(12,36)(14,44)(15,48)(16,18)(17,26)(19,24)(20,29)(22,35)(25,53)(28,54)(31,58)(32,51)(33,34)(38,56)(40,52)(41,47)(42,45)(43,50)(46,63)(49,64)(55,57)(59,62)(60,61);(1,5,6,28,7,24,29,25)(2,10,11,40,12,22,26,38)(3,14,15,49,16,45,50,46)(4,19,20,54,21,23,27,53)(8,31,32,60,33,55,47,59)(9,35,17,52,36,37,39,56)(13,42,43,64,18,44,48,63)(30,57,41,61,34,58,51,62)Polycyclic presentation:
2^9|a^2=i,b^2=egh,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380105);