Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356478 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | D4×#95 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#338 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1831 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)(3,15,19,9,20,32,18,33)(4,21,23,35,25,40,24,10)(5,11,27,37,28,39,29,38)(16,42,45,41,46,30,44,56)(17,36,47,31,48,57,26,54)(22,49,51,60,53,59,52,34)(43,61,62,58,63,55,50,64);(1,3,7,19,8,20,6,18)(2,9,12,15,14,33,13,32)(4,16,24,45,25,46,23,44)(5,26,27,48,28,47,29,17)(10,30,21,41,35,42,40,56)(11,36,38,31,39,57,37,54)(22,50,51,63,53,62,52,43)(34,58,59,55,60,64,49,61);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,34)(12,21)(13,40)(14,35)(15,41)(17,43)(18,44)(19,45)(20,46)(26,50)(27,51)(28,53)(29,52)(31,55)(32,56)(33,42)(36,58)(37,49)(38,59)(39,60)(47,62)(48,63)(54,61)(57,64);(1,5,6,27,8,28,7,29)(2,11,12,37,14,39,13,38)(3,17,19,47,20,48,18,26)(4,22,23,51,25,53,24,52)(9,31,32,57,33,54,15,36)(10,34,21,49,35,60,40,59)(16,43,45,62,46,63,44,50)(30,55,56,64,42,61,41,58)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,g^b=gi,f^c=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263675);