Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467244 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16562 |
| 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 | |
923 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)(3,15)(4,21)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,24)(16,32)(17,43,49,61)(19,35)(20,34)(22,52)(23,51,54,40)(25,39)(26,38)(28,56,53,37)(29,41,31,42)(33,48,55,50)(36,45)(44,62,57,58)(46,59)(47,60,63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,16,26,45,25,46)(5,17,31,50,30,49,29,48)(10,36,21,32,39,52,38,59)(11,33,42,43,27,55,41,61)(23,47,53,44,54,63,28,57)(37,60,51,58,56,64,40,62);(1,4,7,24,8,26,6,25)(2,10,13,21,14,39,12,38)(3,16,19,45,20,46,18,22)(5,28,29,23,30,53,31,54)(9,32,34,52,35,59,15,36)(11,40,41,37,27,51,42,56)(17,47,48,44,49,63,50,57)(33,60,61,58,55,64,43,62);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,37)(12,42)(13,41)(14,27)(15,43)(16,44)(18,50)(19,48)(20,49)(21,51)(22,47)(24,53)(25,28)(26,54)(32,58)(34,61)(35,55)(36,60)(38,40)(39,56)(45,63)(46,57)(52,64)(59,62)Polycyclic presentation:
2^9|a^2=eg,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374441);