Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #446222 |
| 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 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,19,9,20,34,18,35)(4,21,24,10,25,38,31,39)(5,26,29,42,30,41,28,11)(16,32,22,36,47,50,46,43)(17,33,27,40,49,60,48,44)(23,53,56,51,57,37,55,58)(45,63,52,59,54,61,64,62);(1,3,6,18,8,20,7,19)(2,9,13,34,14,35,12,15)(4,22,24,16,25,46,31,47)(5,27,28,49,30,48,29,17)(10,36,21,50,39,43,38,32)(11,40,26,33,42,44,41,60)(23,54,55,64,57,45,56,52)(37,62,58,61,53,59,51,63);(1,4,7,31,8,25,6,24)(2,10,12,38,14,39,13,21)(3,16,18,46,20,47,19,22)(5,23,28,55,30,57,29,56)(9,32,34,36,35,50,15,43)(11,37,26,58,42,53,41,51)(17,45,48,64,49,54,27,52)(33,59,40,61,60,62,44,63);(1,5,8,30)(2,11)(3,17,20,49)(4,23,25,57)(6,29,7,28)(9,33)(10,37)(12,26)(13,41)(14,42)(15,44)(16,45,47,54)(18,48,19,27)(21,51)(22,52,46,64)(24,56,31,55)(32,59)(34,40)(35,60)(36,61)(38,58)(39,53)(43,63)(50,62)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,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353418);