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 | |
4438 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)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,38)(35,39)(36,40);(1,2,5,6)(3,7,4,8)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(10,15)(11,18)(17,20)(19,24)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618718);