Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503916 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #26174 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1773 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
526 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)(3,17)(4,22)(5,27)(6,31)(7,15)(8,14)(9,16)(10,19)(11,25)(12,28)(13,35)(18,39)(20,47)(21,54)(23,48)(24,43)(26,44)(29,51)(30,63)(32,40)(33,38)(34,50)(36,49)(37,53)(41,57)(42,58)(45,59)(46,60)(52,55)(56,62)(61,64);(1,3,7,20,9,21,8,37)(2,10,14,40,16,42,15,41)(4,23,25,5,26,29,18,30)(6,32,34,19,36,57,35,58)(11,12,22,45,39,46,44,38)(13,47,49,17,50,53,31,54)(24,60,61,59,56,28,55,33)(27,62,48,64,63,43,51,52);(1,4,7,25,9,26,8,18)(2,11,14,22,16,39,15,44)(3,5,20,29,21,30,37,23)(6,24,34,61,36,56,35,55)(10,38,40,12,42,45,41,46)(13,43,49,52,50,62,31,64)(17,51,53,27,54,48,47,63)(19,28,57,33,58,60,32,59);(1,5)(2,12)(3,18)(4,20)(6,33)(7,29)(8,23)(9,30)(10,39)(11,42)(13,48)(14,45)(15,38)(16,46)(17,52)(19,56)(21,25)(22,41)(24,58)(26,37)(27,31)(28,35)(32,61)(34,60)(36,59)(40,44)(43,47)(49,63)(50,51)(53,62)(54,64)(55,57)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411112);