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 | |
3643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,8,12)(3,13,7,9)(4,15,14,10)(5,16,11,6)(18,23)(20,29)(24,30)(27,32)(33,34,38,42)(35,43,44,39)(36,47,48,40)(37,41,51,56)(45,52,58,57)(46,55,50,53)(49,60,61,54)(59,63,62,64);(1,3,8,7)(2,9,12,13)(4,11,14,5)(6,15,16,10)(19,26)(24,30)(27,32)(28,31)(35,44)(36,48)(39,43)(40,47)(46,50)(49,61)(53,55)(54,60);(1,4)(2,10)(3,5)(6,13)(7,11)(8,14)(9,16)(12,15)(17,18,22,23)(19,24,26,30)(20,25,29,21)(27,31,32,28)(33,35,38,44)(34,39,42,43)(36,45,48,58)(37,50,51,46)(40,52,47,57)(41,55,56,53)(49,62,61,59)(54,64,60,63);(1,5,8,11)(2,6,12,16)(3,14,7,4)(9,10,13,15)(17,19,22,26)(18,24,23,30)(20,27,29,32)(21,28,25,31)(33,34)(35,39)(36,47)(37,41)(38,42)(40,48)(43,44)(45,57)(46,53)(49,60)(50,55)(51,56)(52,58)(54,61)(59,64)(62,63);(1,6)(2,11)(3,15)(4,13)(5,12)(7,10)(8,16)(9,14)(17,20)(18,25)(19,27)(21,23)(22,29)(24,31)(26,32)(28,30)(33,36,38,48)(34,40,42,47)(35,45,44,58)(37,49,51,61)(39,52,43,57)(41,54,56,60)(46,59,50,62)(53,63,55,64);(1,7,8,3)(2,9,12,13)(4,11,14,5)(6,10,16,15)(17,21,22,25)(18,20,23,29)(19,28,26,31)(24,27,30,32)(33,37,38,51)(34,41,42,56)(35,46,44,50)(36,49,48,61)(39,53,43,55)(40,54,47,60)(45,59,58,62)(52,63,57,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324318);