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 | |
3715 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,24)(19,20,27,32)(21,25,22,26)(28,30,29,31)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(2,7)(3,10)(9,12)(11,16)(18,24)(19,27)(25,26)(28,29)(33,34,38,42)(35,44,43,39)(36,47,50,40)(37,41,52,51)(45,53,48,55)(46,58,57,54)(49,60,62,56)(59,63,61,64);(1,2,6,7)(3,8,10,5)(4,9,14,12)(11,15,16,13)(17,19)(18,20)(21,28)(22,29)(23,27)(24,32)(25,30)(26,31)(33,34)(35,39)(36,40)(37,51)(38,42)(41,52)(43,44)(45,53)(46,58)(47,50)(48,55)(49,60)(54,57)(56,62)(59,64)(61,63);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20)(18,19)(21,30)(22,31)(23,32)(24,27)(25,28)(26,29)(33,35,38,43)(34,39,42,44)(36,48,50,45)(37,46,52,57)(40,55,47,53)(41,54,51,58)(49,61,62,59)(56,64,60,63);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21,23,22)(18,25,24,26)(19,28,27,29)(20,30,32,31)(33,36,38,50)(34,40,42,47)(35,45,43,48)(37,49,52,62)(39,53,44,55)(41,56,51,60)(46,59,57,61)(54,63,58,64);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,21)(18,26,24,25)(19,29,27,28)(20,31,32,30)(33,37)(34,41)(35,46)(36,49)(38,52)(39,54)(40,56)(42,51)(43,57)(44,58)(45,59)(47,60)(48,61)(50,62)(53,63)(55,64)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,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, 8324416);