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 | |
4045 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(17,18)(19,21)(20,28)(22,23)(24,30)(25,26)(27,32)(29,31)(33,34)(35,44)(36,40)(37,45)(38,43)(39,46)(41,47)(42,48);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35)(34,39)(36,45)(37,41)(38,46)(40,48)(42,47)(43,44);(2,8)(4,9)(10,14)(12,16)(18,23)(19,25)(24,28)(27,31)(33,36,38,47)(34,40,43,41)(35,45,46,42)(37,39,48,44);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,29)(31,32)(33,36)(34,41)(35,45)(37,39)(38,47)(40,43)(42,46)(44,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20,22,30)(18,24,23,28)(19,27,25,31)(21,29,26,32)(33,34)(35,39)(36,40)(37,42)(38,43)(41,47)(44,46)(45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,26)(18,25,23,19)(20,29,30,32)(24,31,28,27)(33,37,38,48)(34,42,43,45)(35,41,46,40)(36,39,47,44)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,e^2=h,f^2=gh,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, 8066429);