Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201717 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
981 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)(3,7,13,20)(4,8)(5,9)(6,10)(11,18,26,30)(12,19,28,32)(14,21)(15,22)(16,23)(17,24)(25,29,27,31)(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);(1,3,6,13)(2,7)(4,11,15,26)(5,12,17,28)(8,18)(9,19)(10,20)(14,25,16,27)(21,29)(22,30)(23,31)(24,32)(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);(1,4,6,15)(2,8,10,22)(3,11,13,26)(5,16,17,14)(7,18,20,30)(9,23,24,21)(12,27,28,25)(19,31,32,29)(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);(1,5,6,17)(2,9,10,24)(3,12,13,28)(4,14,15,16)(7,19,20,32)(8,21,22,23)(11,25,26,27)(18,29,30,31)(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=egh,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109051);