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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(2,3)(6,7)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,3,5,2)(4,7,8,6)(10,16)(12,17)(15,22)(18,23)(26,32)(27,35)(28,37)(34,38);(1,4,5,8)(2,6,3,7)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073944);