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 | #49578 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(26,28)(31,33)(32,36)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20);(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,23,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7766482);