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 |
| prime radical | #2676 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4281 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)(10,15)(11,17)(18,21)(19,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24);(1,3,5,4)(2,7,6,8)(10,15)(12,16)(18,21)(20,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,23,24)(18,22,21,20)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=g,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, 7884697);