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 | #44333 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,19)(11,16,14,18)(12,13,17,24)(20,21,22,23)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,20)(13,21)(14,15)(17,22)(18,19)(23,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(11,14)(16,18)(20,22)(21,23)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(1,2,5,4)(3,6,8,7)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,22)(18,20)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,23,24,21)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 7747128);