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 | |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,12,16,20)(13,17,22,23)(18,19,24,21)(26,31)(27,33)(34,39)(35,40);(1,3,5,4)(2,6,8,7)(10,15)(11,16)(17,23)(18,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,11,14,16)(10,12,15,20)(13,18,22,24)(17,19,23,21);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,15)(13,21)(14,20)(17,18)(19,22)(23,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,13,14,22)(10,17,15,23)(11,18,16,24)(12,19,20,21)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=i,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, 7787048);