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 | |
4531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,18)(11,12,16,15)(13,17,23,24)(19,20,21,22)(25,26,30,32)(27,31,34,29)(28,36,37,33)(35,38,40,39);(1,2,5,6)(3,7,4,8)(9,11,14,16)(10,15,18,12)(13,21,23,19)(17,20,24,22)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(1,3,5,4)(2,7,6,8)(11,16)(12,15)(19,21)(20,22)(26,32)(27,34)(28,37)(38,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,21)(23,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,16,18,11)(13,22,23,20)(17,19,24,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,17,18,24)(11,19,16,21)(12,20,15,22)(25,29,30,31)(26,27,32,34)(28,38,37,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882248);