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:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(17,18,22,23)(19,24,26,21)(20,28,30,25)(27,29,32,31)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(2,8)(3,4)(10,15)(11,12)(18,23)(19,26)(25,28)(27,32)(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);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(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);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(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);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(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);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(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=g,b^2=gi,c^2=h,d^2=i,e^2=gi,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, 8324400);