Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535047 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #59220 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,14,8,15,7,33)(3,16,20,39,21,38,19,10)(4,22,34,42,26,54,25,11)(5,27,32,13,6,30,29,12)(17,35,23,44,50,61,49,40)(18,36,52,45,28,37,51,46)(24,43,59,58,31,41,57,53)(47,63,55,60,48,64,56,62);(1,3,7,19,8,21,9,20)(2,10,14,16,15,39,33,38)(4,23,25,17,26,49,34,50)(5,28,29,52,6,18,32,51)(11,40,22,35,42,44,54,61)(12,36,27,45,13,37,30,46)(24,56,57,48,31,55,59,47)(41,63,53,60,43,64,58,62);(1,4,7,25,8,26,9,34)(2,11,14,22,15,42,33,54)(3,17,20,23,21,50,19,49)(5,24,29,57,6,31,32,59)(10,35,38,40,39,61,16,44)(12,41,27,53,13,43,30,58)(18,47,52,55,28,48,51,56)(36,60,46,63,37,62,45,64);(1,5,7,29,8,6,9,32)(2,12,14,27,15,13,33,30)(3,18,19,51,21,28,20,52)(4,24,25,57,26,31,34,59)(10,36,16,45,39,37,38,46)(11,41,22,53,42,43,54,58)(17,47,49,56,50,48,23,55)(35,60,44,64,61,62,40,63);(1,6,7,32,8,5,9,29)(2,13,14,30,15,12,33,27)(3,18,19,51,21,28,20,52)(4,24,25,57,26,31,34,59)(10,37,16,46,39,36,38,45)(11,41,22,53,42,43,54,58)(17,48,49,55,50,47,23,56)(35,60,44,64,61,62,40,63)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=f,e^2=f,f^2=i,g^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,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497674);