Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,6,9)(3,4,11,7)(5,13,14,8)(10,15,16,12)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,39,37)(35,43,44,40)(36,47,50,41)(38,51,53,42)(45,54,48,56)(46,55,52,58)(49,57,62,60)(59,64,61,63);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(35,44)(40,43)(45,48)(46,52)(54,56)(55,58)(59,61)(63,64);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,44)(34,40,37,43)(36,48,50,45)(38,52,53,46)(41,56,47,54)(42,58,51,55)(49,59,62,61)(57,63,60,64);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36,39,50)(34,41,37,47)(35,45,44,48)(38,49,53,62)(40,54,43,56)(42,57,51,60)(46,59,52,61)(55,63,58,64);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,34)(35,43,44,40)(36,41,50,47)(38,51,53,42)(45,54,48,56)(46,58,52,55)(49,57,62,60)(59,63,61,64);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,53)(34,42,37,51)(35,46,44,52)(36,49,50,62)(40,55,43,58)(41,57,47,60)(45,59,48,61)(54,63,56,64)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839418);