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 | #38949 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,3,5,8)(2,6,7,4)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(27,28)(29,38)(31,34)(32,37);(1,3)(2,4)(5,8)(6,7)(12,20)(15,22)(18,24)(21,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109555);