Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #438230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15615 |
| 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 | |
936 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)(3,14)(4,20)(5,25)(6,30)(7,12)(8,13)(9,17)(10,23)(11,29)(15,32)(16,40)(18,35)(19,34)(21,36)(22,49)(24,38)(26,33)(27,41)(28,54)(31,39)(37,56)(42,57)(43,58)(44,59)(45,60)(46,52)(47,51)(48,61)(50,53)(55,64)(62,63);(1,3,7,17,8,19,6,18)(2,9,12,34,13,35,30,14)(4,21,23,44,24,43,31,15)(5,26,28,46,29,45,27,16)(10,36,38,59,39,58,20,32)(11,40,41,33,25,52,54,60)(22,42,50,48,51,63,56,55)(37,62,49,64,53,57,47,61);(1,4,6,23,8,24,7,31)(2,10,12,20,13,39,30,38)(3,15,18,21,19,44,17,43)(5,22,27,50,29,51,28,56)(9,32,34,58,35,59,14,36)(11,37,41,47,25,53,54,49)(16,42,45,48,46,63,26,55)(33,57,52,64,60,62,40,61);(1,5,6,27,8,29,7,28)(2,11,12,41,13,25,30,54)(3,16,17,26,19,46,18,45)(4,22,23,50,24,51,31,56)(9,33,14,40,35,60,34,52)(10,37,20,47,39,53,38,49)(15,42,43,55,44,63,21,48)(32,57,36,61,59,62,58,64)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,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, 345426);