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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,18)(12,17)(13,20)(14,16)(15,21)(19,23)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,21)(10,15,16,18)(12,20,22,19)(13,17,23,24)(25,26,30,31)(27,32,37,40)(28,33,38,36)(29,34,39,35);(1,3,5,4)(2,6,8,7)(10,16)(12,22)(15,18)(19,20)(26,31)(29,39)(32,40)(33,36);(9,12,14,22)(10,17,16,24)(11,20,21,19)(13,15,23,18)(25,27)(26,32)(28,34)(29,36)(30,37)(31,40)(33,39)(35,38);(1,3,5,4)(2,6,8,7)(9,11)(10,18)(12,20)(13,17)(14,21)(15,16)(19,22)(23,24)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,4,5,3)(2,7,8,6)(9,13)(10,19)(11,17)(12,15)(14,23)(16,20)(18,22)(21,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=gh,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, 7908214);