Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395737 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14726 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1797 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)(3,15)(4,21)(5,27,29,38)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28,39,30)(16,42)(17,32,47,59)(19,34)(20,33)(22,31)(23,51,53,35)(24,37)(26,36)(40,45)(41,46,55,48)(43,62,50,60)(44,58)(49,52,56,54)(57,64,61,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,45,26,16,25,44)(5,17,28,46,29,47,30,48)(10,31,36,58,37,42,21,40)(11,32,38,41,39,59,27,55)(23,50,52,64,53,43,54,63)(35,57,56,62,51,61,49,60);(1,4,8,26)(2,10,14,37)(3,16,20,22)(5,23,29,53)(6,25,7,24)(9,31,34,42)(11,35,39,51)(12,21,13,36)(15,40,33,58)(17,43,47,50)(18,45,19,44)(27,56,38,49)(28,54,30,52)(32,57,59,61)(41,60,55,62)(46,64,48,63);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,35)(12,38)(13,27)(14,39)(15,41)(16,43)(18,46)(19,48)(20,47)(21,49)(22,50)(24,52)(25,54)(26,53)(31,57)(33,55)(34,59)(36,56)(37,51)(40,60)(42,61)(44,63)(45,64)(58,62)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302934);