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 | |
3641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2,7,4)(3,6,11,8)(5,9,14,12)(10,13,16,15)(17,18,24,28)(19,29,31,22)(20,30,32,25)(21,27,23,26)(33,34,39,37)(35,43,44,40)(36,47,50,41)(38,51,53,42)(45,54,48,56)(46,55,52,58)(49,57,62,60)(59,64,61,63);(1,2,7,4)(3,8,11,6)(5,12,14,9)(10,13,16,15)(17,19,24,31)(18,22,28,29)(20,27,32,26)(21,25,23,30)(35,44)(40,43)(45,48)(46,52)(54,56)(55,58)(59,61)(63,64);(1,3,7,11)(2,8,4,6)(5,10,14,16)(9,15,12,13)(17,20,24,32)(18,25,28,30)(19,26,31,27)(21,29,23,22)(33,35,39,44)(34,40,37,43)(36,48,50,45)(38,52,53,46)(41,56,47,54)(42,58,51,55)(49,59,62,61)(57,63,60,64);(1,4,7,2)(3,6,11,8)(5,12,14,9)(10,13,16,15)(17,21,24,23)(18,26,28,27)(19,25,31,30)(20,29,32,22)(33,36,39,50)(34,41,37,47)(35,45,44,48)(38,49,53,62)(40,54,43,56)(42,57,51,60)(46,59,52,61)(55,63,58,64);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22)(18,19)(20,21)(23,32)(24,29)(25,26)(27,30)(28,31)(33,37,39,34)(35,43,44,40)(36,41,50,47)(38,51,53,42)(45,54,48,56)(46,58,52,55)(49,57,62,60)(59,63,61,64);(1,6)(2,3)(4,11)(5,13)(7,8)(9,10)(12,16)(14,15)(17,23,24,21)(18,27,28,26)(19,30,31,25)(20,22,32,29)(33,38)(34,42)(35,46)(36,49)(37,51)(39,53)(40,55)(41,57)(43,58)(44,52)(45,59)(47,60)(48,61)(50,62)(54,63)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839326);