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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(2,6)(3,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,2)(3,8)(4,7)(5,6)(9,11,14,12)(10,16,18,15)(13,20,21,19)(17,22,24,23)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(2,6)(3,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,4)(5,7)(6,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9401972);