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 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,4,5,8)(6,14,15,10)(11,16,13,12)(21,29)(24,30)(26,31)(27,32)(33,34,39,37)(35,43,46,40)(36,47,45,41)(38,48,44,42);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(2,9)(4,8)(10,14)(12,16)(18,23)(19,25)(24,30)(26,31)(33,34,39,37)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,5,7,3)(2,4,9,8)(6,13,15,11)(10,12,14,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37,39,34)(35,43,46,40)(36,41,45,47)(38,48,44,42);(1,6,7,15)(2,10,9,14)(3,11,5,13)(4,12,8,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066702);