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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,8,7)(3,11,6,9)(4,14,10,5)(12,13,15,16)(17,18,23,19)(20,27,30,24)(21,25,22,26)(28,32,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,53)(44,54)(45,51)(46,47)(48,55)(49,56)(50,52)(57,58)(59,63)(60,61)(62,64);(1,3,8,6)(2,9,7,11)(4,15,10,12)(5,16,14,13)(18,19)(24,27)(25,26)(31,32)(34,39)(36,46)(40,51)(42,52)(43,57)(48,60)(54,56)(59,62);(1,4)(2,5)(3,12)(6,15)(7,14)(8,10)(9,13)(11,16)(17,19,23,18)(20,27,30,24)(21,26,22,25)(28,32,29,31)(33,35,38,45)(34,40,39,51)(36,43,46,57)(37,49,50,44)(41,53,47,58)(42,56,52,54)(48,62,60,59)(55,64,61,63);(1,5,8,14)(2,10,7,4)(3,13,6,16)(9,15,11,12)(17,20,23,30)(18,24,19,27)(21,28,22,29)(25,31,26,32)(33,34)(35,40)(36,47)(37,42)(38,39)(41,46)(43,58)(44,54)(45,51)(48,61)(49,56)(50,52)(53,57)(55,60)(59,64)(62,63);(1,6,8,3)(2,11,7,9)(4,15,10,12)(5,16,14,13)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)(33,36)(34,41)(35,43)(37,48)(38,46)(39,47)(40,53)(42,55)(44,59)(45,57)(49,62)(50,60)(51,58)(52,61)(54,63)(56,64);(1,7,8,2)(3,11,6,9)(4,14,10,5)(12,16,15,13)(17,22,23,21)(18,26,19,25)(20,29,30,28)(24,32,27,31)(33,37,38,50)(34,42,39,52)(35,44,45,49)(36,48,46,60)(40,54,51,56)(41,55,47,61)(43,59,57,62)(53,63,58,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=i,e^2=i,f^2=g,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, 8324193);