Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14913 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,26,37)(5,27,29,11)(6,13,7,12)(9,19,34,18)(10,24,38,25)(16,31,22,35)(17,46,48,42)(23,50,53,60)(28,39,30,40)(32,47,55,49)(36,52,56,54)(41,44,58,45)(43,57,51,59)(61,62,64,63);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,45,26,16,25,44)(5,17,30,49,29,48,28,47)(10,35,37,58,38,31,21,41)(11,32,40,46,27,55,39,42)(23,51,52,63,53,43,54,62)(36,59,60,64,56,57,50,61);(1,4,6,24,8,26,7,25)(2,10,12,37,14,38,13,21)(3,16,18,44,20,22,19,45)(5,23,28,52,29,53,30,54)(9,31,15,41,34,35,33,58)(11,36,39,60,27,56,40,50)(17,43,47,62,48,51,49,63)(32,57,42,61,55,59,46,64);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,39)(13,40)(14,27)(15,42)(16,43)(18,47)(19,49)(20,48)(21,50)(22,51)(24,52)(25,54)(26,53)(31,57)(33,46)(34,55)(35,59)(37,60)(38,56)(41,61)(44,62)(45,63)(58,64)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312693);