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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,11)(14,16)(18,23)(19,26)(20,28)(29,31)(33,34)(35,40)(36,42)(37,41)(38,43)(39,46)(44,47)(45,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,47)(42,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36,38,37)(34,41,43,42)(35,44,46,45)(39,47,40,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,37,38,36)(34,42,43,41)(35,45,46,44)(39,48,40,47)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026847);