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 | #36138 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4461 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)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,31,30)(26,32,36,35)(28,39,37,29)(33,34,38,40);(1,3)(2,4)(5,8)(6,7)(25,28)(26,33)(27,29)(30,39)(31,37)(32,40)(34,35)(36,38);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,29,31,39)(26,34,36,40)(27,37,30,28)(32,33,35,38);(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,30,31,27)(26,35,36,32)(28,39,37,29)(33,34,38,40);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,26)(27,32)(28,33)(29,40)(30,35)(31,36)(34,39)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345368);