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 | #52297 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4364 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)(9,10)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24)(25,26,30,31)(27,28,33,32)(29,34,38,39)(35,36,40,37);(2,6)(4,7)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(26,31)(27,33)(34,39)(35,40);(1,3)(2,7)(4,6)(5,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,4,5,7)(2,3,6,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,20)(22,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^2=i,f^2=i,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, 8003818);