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 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,12)(11,13,17,16)(14,20,22,18)(19,23,24,21);(1,2,4,5)(3,6,7,8)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(25,26,30,31)(27,32,36,39)(28,29,33,38)(34,40,37,35);(1,3,4,7)(2,6,5,8)(10,12)(11,17)(14,22)(21,23)(26,31)(29,38)(32,39)(35,40);(9,12,15,10)(11,13,17,16)(14,20,22,18)(19,21,24,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(9,13)(10,17)(11,12)(14,21)(15,16)(18,24)(19,20)(22,23)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(9,14,15,22)(10,18,12,20)(11,19,17,24)(13,21,16,23)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919218);