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 | |
4615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(26,28)(31,34)(32,36)(38,40);(1,3,5,8)(2,6,7,4)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,33)(31,34)(35,39)(38,40);(1,3)(2,4)(5,8)(6,7)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=h,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044400);