Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,12,16,23)(4,17)(5,10)(6,21)(7,11)(8,15,24,14)(9,19)(13,28,31,22)(18,26)(20,25)(27,29,32,30)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,59,63)(36,46,48,56,47,40,49,54)(43,51,58,62,57,61,60,64);(1,3,5,14,7,16,6,15)(2,8,10,23,11,24,21,12)(4,13,19,30,20,31,18,29)(9,22,25,32,26,28,17,27)(33,35,37,45,38,44,50,59)(34,39,42,53,41,52,55,63)(36,43,49,60,47,57,48,58)(40,51,56,64,46,61,54,62);(1,4,7,20)(2,9,11,26)(3,13,16,31)(5,19,6,18)(8,22,24,28)(10,25,21,17)(12,27,23,32)(14,30,15,29)(33,36)(34,40)(35,43)(37,49)(38,47)(39,51)(41,46)(42,56)(44,57)(45,60)(48,50)(52,61)(53,64)(54,55)(58,59)(62,63);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,20,19)(8,23,24,12)(9,17,26,25)(13,29,31,30)(22,27,28,32)(34,41)(39,52)(40,46)(42,55)(51,61)(53,63)(54,56)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102603);