Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441801 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15695 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
807 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)(3,13,17,8)(4,18,21,46)(5,22)(6,12)(7,11)(9,20,33,38)(10,24)(14,37)(15,34,43,28)(16,36,25,29)(19,47,49,45)(23,52)(26,31)(27,40)(30,50)(32,58,51,48)(35,54)(39,60)(41,44)(42,59,53,56)(55,61)(57,64)(62,63);(1,3,6,25,7,17,26,16)(2,8,12,36,11,13,31,29)(4,14,38,40,21,41,20,50)(5,15,23,53,24,43,54,42)(9,30,46,37,33,27,18,44)(10,34,35,56,22,28,52,59)(19,39,58,61,49,62,48,64)(32,55,47,63,51,57,45,60);(1,4,6,20,7,21,26,38)(2,9,31,46,11,33,12,18)(3,14,16,40,17,41,25,50)(5,19,23,48,24,49,54,58)(8,27,36,37,13,30,29,44)(10,32,52,47,22,51,35,45)(15,39,42,61,43,62,53,64)(28,55,59,60,34,57,56,63);(1,5,7,24)(2,10)(3,15,17,43)(4,19,21,49)(6,23,26,54)(8,28)(9,32)(11,22)(12,35)(13,34)(14,39,41,62)(16,42,25,53)(18,45)(20,48,38,58)(27,55)(29,56)(30,57)(31,52)(33,51)(36,59)(37,60)(40,61,50,64)(44,63)(46,47)Polycyclic presentation:
2^9|a^2=eg,b^2=efg,c^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348998);