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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(10,15)(11,17)(18,23)(19,24)(25,26)(27,33)(28,32)(29,37)(30,34)(31,39)(35,38)(36,40);(2,6)(4,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27)(26,29)(28,36)(30,37)(31,40)(32,38)(33,34)(35,39);(1,3)(2,7)(4,6)(5,8)(25,28)(26,31)(27,35)(29,38)(30,39)(32,34)(33,40)(36,37);(1,4,5,7)(2,3,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,29,30,33)(26,27,34,37)(28,38,39,40)(31,35,32,36);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,28)(26,32)(27,36)(29,38)(30,39)(31,34)(33,40)(35,37);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,27)(26,33)(28,35)(29,34)(30,37)(31,40)(32,38)(36,39)Polycyclic presentation:
2^9|a^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004070);