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 | |
4253 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,25)(20,21)(22,26)(23,29)(24,30)(27,31)(28,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,7)(3,9)(8,14)(10,15)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,22,30)(18,24,26,21)(19,27,29,32)(23,31,25,28)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,24)(18,20,26,30)(19,28,29,31)(23,27,25,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,19)(18,25)(20,27)(21,31)(22,29)(23,26)(24,28)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013763);