Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479550 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17214 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
628 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)(3,15,19,9)(4,20)(5,26)(6,14)(7,13)(8,12)(10,24)(11,28)(16,46,48,35)(17,45,49,33)(18,34,31,43)(21,44,51,32)(22,52)(23,37)(25,38)(27,58,50,39)(29,41)(30,42)(36,55)(40,56)(47,61,54,63)(53,59,64,60)(57,62);(1,3,6,18,8,19,7,31)(2,9,14,43,12,15,13,34)(4,21,25,16,23,51,24,48)(5,27,30,49,28,50,29,17)(10,35,20,44,38,46,37,32)(11,39,41,45,26,58,42,33)(22,54,55,64,57,47,56,53)(36,59,62,63,40,60,52,61);(1,4,6,25,8,23,7,24)(2,10,14,20,12,38,13,37)(3,16,18,51,19,48,31,21)(5,22,30,55,28,57,29,56)(9,32,43,35,15,44,34,46)(11,40,41,52,26,36,42,62)(17,47,27,53,49,54,50,64)(33,60,39,61,45,59,58,63);(1,5,7,30,8,28,6,29)(2,11,14,42,12,26,13,41)(3,17,31,27,19,49,18,50)(4,22,25,56,23,57,24,55)(9,33,43,58,15,45,34,39)(10,36,37,62,38,40,20,52)(16,47,51,64,48,54,21,53)(32,59,46,63,44,60,35,61)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=ef,d^2=fi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386747);