Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15468 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 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)(3,13,16,8)(4,18)(5,23)(6,26)(7,11)(9,21)(10,24)(12,29)(14,30,19,33)(15,42,43,55)(17,32,27,38)(20,50)(22,35)(25,39)(28,37)(31,44,36,45)(34,54)(40,61,48,64)(41,60,49,58)(46,52)(47,53)(51,56)(57,63,59,62);(1,3)(2,8)(4,19,21,14)(5,15)(6,27)(7,16)(9,33,18,30)(10,36)(11,13)(12,38)(17,29)(20,48,51,40)(22,41,53,49)(23,55)(24,45)(25,43)(26,32)(28,44)(31,37)(34,57,46,59)(35,58,47,60)(39,42)(50,61,56,64)(52,62,54,63);(1,4)(2,9,11,18)(3,14)(5,20)(6,22)(7,21)(8,30,13,33)(10,34,37,46)(12,35,26,47)(15,40)(16,19)(17,41)(23,56,39,50)(24,52)(25,51)(27,49)(28,54)(29,53)(31,57,36,59)(32,58,38,60)(42,64,55,61)(43,48)(44,62)(45,63);(1,5,7,25)(2,10,11,37)(3,15,16,43)(4,20,21,51)(6,28,29,24)(8,31,13,36)(9,34,18,46)(12,39,26,23)(14,40,19,48)(17,45,27,44)(22,54,53,52)(30,57,33,59)(32,42,38,55)(35,50,47,56)(41,63,49,62)(58,64,60,61)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340090);