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 | |
4379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,13)(14,17)(15,18)(16,20)(21,24)(22,23)(27,29)(32,34)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,17)(13,21)(14,23)(15,20)(18,19)(22,24)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(11,20)(14,21)(16,19)(17,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(1,4,5,2)(3,7,8,6)(9,12,15,22)(10,13,18,23)(11,17,20,24)(14,16,21,19)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,13)(10,12)(11,21)(14,20)(15,23)(16,17)(18,22)(19,24)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(9,14,15,21)(10,17,18,24)(11,13,20,23)(12,16,22,19)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7749612);