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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(18,22)(24,26)(25,28)(31,32)(34,40)(35,45)(37,42)(43,46);(2,9)(3,12)(4,11)(10,14)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,21,23,19)(18,26,22,24)(20,29,30,27)(25,32,28,31)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662230);