Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #362481 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2293 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13921 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
675 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)(3,13,17,39)(4,9,21,34)(5,23,26,52)(6,11)(7,12)(8,16,29,44)(10,25,35,53)(14,37)(15,43)(18,27)(19,48)(20,32,22,33)(24,36)(28,46)(30,42)(31,51)(38,60)(40,62,64,54)(41,55)(45,61)(47,59,63,56)(49,58)(50,57);(1,3)(2,8,12,29)(4,18,21,41)(5,15)(6,16)(7,17)(9,30)(10,28,35,43)(11,13,36,39)(14,22,42,20)(19,47,50,63)(23,38,52,61)(24,44)(25,45)(26,46)(27,32)(31,56)(33,55)(34,37)(40,51,64,49)(48,54)(53,60)(57,62)(58,59);(1,4,6,20,7,21,24,22)(2,9,11,32,12,34,36,33)(3,14,16,41,17,42,44,18)(5,19,25,49,26,50,53,51)(8,27,13,37,29,55,39,30)(10,31,23,48,35,58,52,57)(15,40,45,63,46,64,60,47)(28,54,38,59,43,62,61,56);(1,5,24,53,7,26,6,25)(2,10,11,23,12,35,36,52)(3,15,44,60,17,46,16,45)(4,19,22,51,21,50,20,49)(8,28,13,38,29,43,39,61)(9,31,32,48,34,58,33,57)(14,40,18,47,42,64,41,63)(27,54,37,59,55,62,30,56)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269883);