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 | #45255 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(3,7)(6,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(26,28)(31,33)(32,36)(39,40);(1,2,5,4)(3,6,7,8)(10,15)(13,16)(17,20)(21,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,3,5,7)(2,6,4,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,26,30,28)(27,31,35,33)(29,32,38,36)(34,39,37,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|c^2=gh,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8149509);