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 | #46799 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,30,32)(27,31,28,33)(29,37,38,34)(35,40,36,39);(1,2)(3,6)(4,7)(5,8)(25,27)(26,31)(28,30)(29,35)(32,33)(34,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(10,15)(11,18)(17,20)(19,24)(26,32)(27,28)(34,37)(35,36);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,30,27)(26,33,32,31)(29,36,38,35)(34,40,37,39);(1,3,5,4)(2,6,8,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38);(1,4,5,3)(2,7,8,6)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29,30,38)(26,34,32,37)(27,35,28,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622175);