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 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,12,17,16)(13,18,21,22)(19,20,24,23)(25,26)(27,33)(28,31)(29,37)(30,32)(34,38)(35,39)(36,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(18,22)(19,24)(25,27,30,28)(26,31,32,33)(29,35,38,36)(34,39,37,40);(2,6)(3,7)(13,21)(18,22)(19,24)(20,23)(26,32)(31,33)(34,37)(39,40);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,30,27)(26,33,32,31)(29,36,38,35)(34,40,37,39);(1,4)(2,3)(5,8)(6,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(1,3)(2,4)(5,7)(6,8)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104480);