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 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(5,13)(8,15)(10,16)(12,14)(18,22)(24,26)(25,28)(31,32)(34,38)(39,41)(40,44)(47,48);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21,23,19)(18,26,22,24)(20,27,30,29)(25,31,28,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202804);