Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459145 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2751 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16333 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
904 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)(3,15,18,35,20,34,19,9)(4,21,24,39,25,38,31,10)(5,26,28,11,30,41,29,42)(16,45,47,43,48,32,22,36)(17,49,51,62,53,33,52,44)(23,55,27,58,57,37,56,40)(46,61,50,64,59,63,54,60);(1,3,6,19,8,20,7,18)(2,9,12,34,14,35,13,15)(4,22,24,48,25,47,31,16)(5,17,28,52,30,53,29,51)(10,36,21,32,39,43,38,45)(11,33,41,62,42,49,26,44)(23,54,27,59,57,50,56,46)(37,61,40,60,55,63,58,64);(1,4)(2,10)(3,16)(5,27)(6,31)(7,24)(8,25)(9,32)(11,40)(12,38)(13,21)(14,39)(15,43)(17,50)(18,22)(19,47)(20,48)(23,28)(26,55)(29,57)(30,56)(33,61)(34,36)(35,45)(37,41)(42,58)(44,60)(46,51)(49,63)(52,59)(53,54)(62,64);(1,5,6,28,8,30,7,29)(2,11,12,41,14,42,13,26)(3,17,19,52,20,53,18,51)(4,23,24,27,25,57,31,56)(9,33,34,62,35,49,15,44)(10,37,21,40,39,55,38,58)(16,46,22,54,48,59,47,50)(32,60,43,63,45,64,36,61)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366341);