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 | |
4254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,8,12,7)(4,9,15,14)(5,10,16,13)(20,28)(24,29)(27,32)(30,31)(34,40)(36,43)(37,44)(41,48);(3,12)(4,15)(7,8)(9,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,34,39,40)(35,41,45,48)(36,38,43,47)(37,46,44,42);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,16)(10,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,35,39,45)(34,41,40,48)(36,44,43,37)(38,42,47,46);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36)(34,38)(35,44)(37,45)(39,43)(40,47)(41,42)(46,48);(1,4)(2,9)(3,13)(5,8)(6,15)(7,16)(10,12)(11,14)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,37,39,44)(34,42,40,46)(35,36,45,43)(38,41,47,48);(1,5)(2,10)(3,14)(4,8)(6,16)(7,15)(9,12)(11,13)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,38)(34,43)(35,42)(36,40)(37,41)(39,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413493);