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 | |
3631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(19,27)(20,29)(23,30)(24,32)(33,34)(35,43)(36,41)(37,39)(38,42)(40,46)(44,57)(45,56)(47,55)(48,49)(50,52)(51,54)(53,59)(58,64)(60,63)(61,62);(2,6)(7,9)(8,13)(14,15)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(34,37)(40,43)(41,49)(42,50)(53,57)(54,56)(55,62)(63,64);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,27)(18,23,26,30)(20,28,29,21)(24,31,32,25)(33,35)(34,40)(36,44)(37,43)(38,51)(39,46)(41,53)(42,56)(45,52)(47,60)(48,59)(49,57)(50,54)(55,64)(58,61)(62,63);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,39,48)(34,41,37,49)(35,44,46,59)(38,47,52,61)(40,53,43,57)(42,55,50,62)(45,58,51,60)(54,63,56,64);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,28)(21,27)(22,29)(23,31)(25,30)(26,32)(33,37,39,34)(35,43,46,40)(36,41,48,49)(38,50,52,42)(44,53,59,57)(45,56,51,54)(47,55,61,62)(58,63,60,64);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,28)(18,25,26,31)(19,20,27,29)(23,24,30,32)(33,38,39,52)(34,42,37,50)(35,45,46,51)(36,47,48,61)(40,54,43,56)(41,55,49,62)(44,58,59,60)(53,63,57,64)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,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, 8324287);