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 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,14,15,10)(11,16,12,13)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,44)(35,42,37,40)(36,41,38,43)(45,48,46,47);(1,3)(2,5)(4,7)(6,11)(8,9)(10,13)(12,15)(14,16)(19,25)(22,28)(26,31)(29,32)(33,35,39,37)(34,40,44,42)(36,45,38,46)(41,47,43,48);(2,8)(3,4)(10,14)(11,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,38)(34,41,44,43)(35,45,37,46)(40,47,42,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,37,39,35)(34,42,44,40)(36,45,38,46)(41,47,43,48);(1,5,7,9)(2,3,8,4)(6,13,15,16)(10,11,14,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,38,39,36)(34,43,44,41)(35,45,37,46)(40,47,42,48);(1,6,7,15)(2,10,8,14)(3,11,4,12)(5,13,9,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021616);