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 | #59013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4588 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26)(27,35)(28,33)(29,36)(30,34)(31,32)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,36)(26,29,32,35)(28,30,37,40)(33,38,39,34);(1,3,5,4)(2,6,8,7)(10,12)(16,17)(18,20)(23,24)(26,32)(27,36)(28,37)(34,38);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,37)(26,33,32,39)(27,30,36,40)(29,38,35,34);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,35)(26,27,32,36)(28,38,37,34)(30,39,40,33);(1,4,5,3)(2,7,8,6)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,34)(27,37)(28,36)(29,33)(31,40)(32,38)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7787060);