Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446220 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15835 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,18,35,19,34,31,9)(4,20,24,39,25,38,23,10)(5,26,28,42,29,11,30,41)(16,32,21,36,45,58,50,57)(17,43,46,59,47,40,27,33)(22,48,51,54,52,61,53,37)(44,63,49,62,64,56,55,60);(1,3,6,18,8,19,7,31)(2,9,13,34,14,35,12,15)(4,21,23,45,25,50,24,16)(5,27,28,47,29,46,30,17)(10,36,20,32,39,57,38,58)(11,40,42,33,26,43,41,59)(22,44,51,55,52,64,53,49)(37,56,61,60,54,63,48,62);(1,4,6,23,8,25,7,24)(2,10,13,20,14,39,12,38)(3,16,31,50,19,45,18,21)(5,22,30,53,29,52,28,51)(9,32,15,36,35,58,34,57)(11,37,41,48,26,54,42,61)(17,44,27,55,47,64,46,49)(33,56,43,60,59,63,40,62);(1,5,8,29)(2,11)(3,17,19,47)(4,22,25,52)(6,30,7,28)(9,33)(10,37)(12,42)(13,41)(14,26)(15,43)(16,44,45,64)(18,46,31,27)(20,48)(21,49,50,55)(23,53,24,51)(32,56)(34,40)(35,59)(36,60)(38,61)(39,54)(57,62)(58,63)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353416);