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 | |
4129 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)(3,7)(4,9)(5,13)(6,8)(10,15)(11,12)(14,16)(17,18,21,24)(19,25,27,22)(20,23,30,29)(26,32,28,31)(33,34,40,43)(35,39,38,41)(36,42,37,47)(44,46,45,48);(1,3,6,9)(2,7,8,4)(5,11,15,16)(10,14,13,12)(17,19,21,27)(18,22,24,25)(20,28,30,26)(23,32,29,31)(33,35,40,38)(34,41,43,39)(36,44,37,45)(42,48,47,46);(2,8)(3,9)(10,13)(11,16)(17,19,21,27)(18,22,24,25)(20,26,30,28)(23,31,29,32)(33,36,40,37)(34,42,43,47)(35,44,38,45)(39,46,41,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,19)(18,22)(20,28)(21,27)(23,32)(24,25)(26,30)(29,31)(33,37,40,36)(34,42,43,47)(35,44,38,45)(39,48,41,46);(1,4)(2,9)(3,8)(5,14)(6,7)(10,11)(12,15)(13,16)(17,20,21,30)(18,23,24,29)(19,26,27,28)(22,31,25,32)(33,38,40,35)(34,41,43,39)(36,45,37,44)(42,48,47,46);(1,5,6,15)(2,10,8,13)(3,11,9,16)(4,12,7,14)(17,18)(19,22)(20,29)(21,24)(23,30)(25,27)(26,32)(28,31)(33,39,40,41)(34,38,43,35)(36,46,37,48)(42,45,47,44)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154877);