Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6447057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #34943 |
| prime radical | #28250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,14,9,15,8,33)(3,16,20,10,21,36,19,37)(4,22,26,41,28,56,27,11)(5,12,6,13,32,44,31,45)(17,48,23,46,52,34,51,38)(18,47,29,35,54,42,53,62)(24,55,59,43,30,40,25,39)(49,61,57,64,50,60,58,63);(1,3,8,20,9,21,7,19)(2,10,14,16,15,37,33,36)(4,23,27,52,28,51,26,17)(5,29,31,54,32,53,6,18)(11,38,22,34,41,46,56,48)(12,42,13,35,44,47,45,62)(24,50,25,58,30,49,59,57)(39,61,55,63,43,60,40,64);(1,4)(2,11,15,41)(3,17,21,52)(5,30)(6,25)(7,26)(8,27)(9,28)(10,34)(12,43,44,39)(13,40,45,55)(14,22,33,56)(16,46)(18,50,54,49)(19,51,20,23)(24,32)(29,58,53,57)(31,59)(35,61)(36,38)(37,48)(42,64)(47,63)(60,62);(1,5,7,6,9,32,8,31)(2,12,14,13,15,44,33,45)(3,18,20,29,21,54,19,53)(4,24,26,59,28,30,27,25)(10,35,36,42,37,62,16,47)(11,39,22,55,41,43,56,40)(17,49,23,57,52,50,51,58)(34,60,38,63,48,61,46,64);(1,6,7,32,9,31,8,5)(2,13,14,44,15,45,33,12)(3,18,19,53,21,54,20,29)(4,25,26,24,28,59,27,30)(10,35,16,47,37,62,36,42)(11,40,22,39,41,55,56,43)(17,50,51,57,52,49,23,58)(34,61,46,63,48,60,38,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375320);