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 | |
3709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(2,6)(7,12)(8,13)(14,16)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(33,34,39,40)(35,41,47,55)(36,37,42,50)(38,43,54,56)(44,45,48,51)(46,57,53,59)(49,52,58,63)(60,61,62,64);(4,11)(8,13)(9,15)(14,16)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(34,40)(37,50)(41,55)(43,56)(45,51)(52,63)(57,59)(61,64);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,35,39,47)(34,41,40,55)(36,48,42,44)(37,51,50,45)(38,53,54,46)(43,59,56,57)(49,60,58,62)(52,61,63,64);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,30,25,32)(33,36)(34,37)(35,44)(38,49)(39,42)(40,50)(41,45)(43,52)(46,60)(47,48)(51,55)(53,62)(54,58)(56,63)(57,61)(59,64);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,37)(34,42)(35,45)(36,40)(38,52)(39,50)(41,48)(43,58)(44,55)(46,61)(47,51)(49,56)(53,64)(54,63)(57,62)(59,60);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,38,39,54)(34,43,40,56)(35,46,47,53)(36,49,42,58)(37,52,50,63)(41,57,55,59)(44,60,48,62)(45,61,51,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=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, 8961186);