Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535051 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #59041 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,22,35)(4,13,26,39)(5,27,32,43)(6,11,25,40)(7,15,8,14)(10,21,36,20)(12,30,44,31)(18,45,48,33)(19,49,50,34)(23,37,47,58)(24,53,56,38)(28,41,51,59)(29,42,55,62)(46,63,54,60)(52,61,64,57);(1,3,7,20,9,22,8,21)(2,10,14,17,16,36,15,35)(4,23,25,48,26,47,6,18)(5,28,30,50,32,51,31,19)(11,37,39,45,40,58,13,33)(12,41,43,49,44,59,27,34)(24,54,55,64,56,46,29,52)(38,61,62,63,53,57,42,60);(1,4,7,25,9,26,8,6)(2,11,14,39,16,40,15,13)(3,18,21,47,22,48,20,23)(5,24,30,55,32,56,31,29)(10,33,35,58,36,45,17,37)(12,38,43,62,44,53,27,42)(19,46,51,64,50,54,28,52)(34,57,59,63,49,61,41,60);(1,5,7,30,9,32,8,31)(2,12,14,43,16,44,15,27)(3,19,20,28,22,50,21,51)(4,24,25,55,26,56,6,29)(10,34,17,41,36,49,35,59)(11,38,39,62,40,53,13,42)(18,46,23,52,48,54,47,64)(33,57,37,60,45,61,58,63);(1,6,8,26,9,25,7,4)(2,13,15,40,16,39,14,11)(3,18,21,47,22,48,20,23)(5,29,31,56,32,55,30,24)(10,33,35,58,36,45,17,37)(12,42,27,53,44,62,43,38)(19,46,51,64,50,54,28,52)(34,57,59,63,49,61,41,60)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7442250);