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 | |
3711 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)(3,12,6,9)(4,11,5,10)(13,15,14,16)(17,18,23,26)(19,21,25,22)(20,28,31,24)(27,30,32,29)(33,34,38,39)(35,43,46,40)(36,41,49,47)(37,42,51,52)(44,57,59,53)(45,56,50,54)(48,55,62,60)(58,64,61,63);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,19,23,25)(18,22,26,21)(20,27,31,32)(24,30,28,29)(34,39)(40,43)(41,47)(42,52)(53,57)(54,56)(55,60)(63,64);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,35,38,46)(34,40,39,43)(36,44,49,59)(37,50,51,45)(41,53,47,57)(42,56,52,54)(48,61,62,58)(55,64,60,63);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,20,23,31)(18,24,26,28)(19,27,25,32)(21,29,22,30)(33,36)(34,41)(35,44)(37,48)(38,49)(39,47)(40,53)(42,55)(43,57)(45,58)(46,59)(50,61)(51,62)(52,60)(54,63)(56,64);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,21,23,22)(18,25,26,19)(20,29,31,30)(24,32,28,27)(33,34)(35,40)(36,47)(37,42)(38,39)(41,49)(43,46)(44,57)(45,54)(48,60)(50,56)(51,52)(53,59)(55,62)(58,64)(61,63);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,22,23,21)(18,19,26,25)(20,30,31,29)(24,27,28,32)(33,37,38,51)(34,42,39,52)(35,45,46,50)(36,48,49,62)(40,54,43,56)(41,55,47,60)(44,58,59,61)(53,63,57,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,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, 8324390);