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 | #37334 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(12,20)(16,23)(19,24)(21,22)(25,26,32,30)(27,36,39,33)(28,40,34,29)(31,35,38,37);(3,4)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,33)(28,37)(29,31)(30,36)(32,39)(34,35)(38,40);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,28)(26,29)(27,37)(30,40)(31,33)(32,34)(35,39)(36,38);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,32,40)(26,34,30,28)(27,38,39,31)(33,37,36,35);(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,30,32,26)(27,36,39,33)(28,40,34,29)(31,37,38,35);(1,5,6,2)(3,8,4,7)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,31,32,38)(26,35,30,37)(27,29,39,40)(28,33,34,36)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=h,e^2=hi,f^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, 8366375);