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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4521 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)(3,8)(4,6)(5,7)(12,20)(16,23)(19,24)(21,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(27,38)(28,39)(32,37)(33,40);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,31,38)(26,32,36,37)(28,35,39,29)(30,33,34,40);(2,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,34)(36,40);(1,4,5,3)(2,8,7,6)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,29,31,35)(26,34,36,30)(27,28,38,39)(32,40,37,33);(1,2)(3,6)(4,8)(5,7)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,30,31,34)(26,35,36,29)(27,33,38,40)(28,32,39,37)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=gi,f^2=i,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, 8365615);