Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477920 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17126 |
| 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 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,28)(3,14,17,34,18,9,29,41)(4,19,30,36,21,49,22,10)(5,23,31,54,26,11,27,38)(15,42,44,39,45,47,59,32)(16,33,24,37,46,40,55,52)(20,48,58,53,25,56,51,35)(43,62,50,61,64,63,57,60);(1,3,8,29,6,18,7,17)(2,9,28,41,13,14,12,34)(4,15,30,59,21,45,22,44)(5,24,31,16,26,55,27,46)(10,32,49,42,36,39,19,47)(11,37,54,40,23,52,38,33)(20,50,58,43,25,57,51,64)(35,62,56,61,53,63,48,60);(1,4,7,22,6,21,8,30)(2,10,12,19,13,36,28,49)(3,15,17,44,18,45,29,59)(5,25,27,58,26,20,31,51)(9,32,34,47,14,39,41,42)(11,35,38,48,23,53,54,56)(16,43,24,50,46,64,55,57)(33,61,52,62,40,60,37,63);(1,5,6,26)(2,11)(3,16)(4,20,21,25)(7,27,8,31)(9,33,14,40)(10,35)(12,38)(13,23)(15,43)(17,24)(18,46)(19,48)(22,51,30,58)(28,54)(29,55)(32,60,39,61)(34,52,41,37)(36,53)(42,62,47,63)(44,50)(45,64)(49,56)(57,59)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,d^c=de,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385117);