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 | #46103 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4177 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)(9,10)(11,17)(12,18)(13,20)(14,15)(16,21)(19,23)(22,24)(26,30)(28,36)(31,33)(34,37);(1,3,5,8)(2,6,4,7)(10,15)(12,22)(16,17)(19,20)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(9,11,14,21)(10,16,15,17)(12,20,22,19)(13,18,23,24)(25,26,29,30)(27,31,35,33)(28,32,36,38)(34,39,37,40);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,20)(13,18)(14,21)(15,16)(19,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,18,15,24)(11,20,21,19)(13,16,23,17)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,36)(38,39);(1,4,5,2)(3,7,8,6)(9,13)(10,19)(11,18)(12,16)(14,23)(15,20)(17,22)(21,24)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7965135);