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 | #50709 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4293 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)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(26,28)(31,33)(32,36)(38,39);(10,15)(13,17)(16,23)(19,24)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,4,5,2)(3,7,8,6)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7706748);