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 | #44882 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4317 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26)(27,34)(28,33)(29,30)(31,32)(35,40)(36,37)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(26,32)(27,29)(33,39)(35,36);(11,20)(13,21)(15,19)(17,24)(25,27,31,29)(26,30,32,34)(28,36,38,35)(33,40,39,37);(9,12)(10,16)(11,21)(13,20)(14,23)(15,17)(18,22)(19,24)(25,28,31,38)(26,33,32,39)(27,35,29,36)(30,37,34,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29,31,27)(26,30,32,34)(28,35,38,36)(33,40,39,37);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,12,20,23)(15,22,19,16)(25,30)(26,27)(28,37)(29,32)(31,34)(33,35)(36,39)(38,40)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=gh,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, 7919196);