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 | |
4145 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)(3,8,4,10)(5,13,15,11)(6,14,16,12)(21,29)(24,30)(26,31)(27,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,9)(3,4)(11,13)(12,14)(18,23)(19,25)(24,30)(26,31)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=i,e^2=h,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, 8066775);