Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423649 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15315 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1514 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)(3,14)(4,20)(5,25)(6,30)(7,12)(8,13)(9,17)(10,23)(11,29)(15,32)(16,40)(18,35)(19,34)(21,36)(22,51)(24,38)(26,33)(27,41)(28,57)(31,39)(37,58)(42,44)(43,59)(45,48)(46,60)(47,55)(49,54)(50,61)(52,64)(53,56)(62,63);(1,3)(2,9)(4,21)(5,26)(6,18)(7,17)(8,19)(10,36)(11,40)(12,34)(13,35)(14,30)(15,23)(16,27)(20,48)(22,52)(24,44)(25,55)(28,47)(29,46)(31,45)(32,38)(33,41)(37,59)(39,42)(43,54)(49,61)(50,53)(51,62)(56,64)(57,60)(58,63);(1,4,6,23,8,24,7,31)(2,10,12,20,13,39,30,38)(3,15,18,44,19,45,17,21)(5,22,27,53,29,54,28,58)(9,32,34,36,35,48,14,42)(11,37,41,49,25,56,57,51)(16,43,46,63,47,52,26,50)(33,59,55,61,60,64,40,62);(1,5,6,27,8,29,7,28)(2,11,12,41,13,25,30,57)(3,16,17,26,19,47,18,46)(4,22,23,53,24,54,31,58)(9,33,14,40,35,60,34,55)(10,37,20,49,39,56,38,51)(15,43,21,50,45,52,44,63)(32,59,42,62,48,64,36,61)Polycyclic presentation:
2^9|c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330845);