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 | |
4545 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)(3,6)(4,7)(5,8)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,36)(28,29)(30,40)(31,33)(32,37)(34,39)(35,38);(1,3)(2,4)(5,7)(6,8)(9,11,14,13)(10,15,16,18)(12,20,21,19)(17,24,23,22)(25,27,31,37)(26,32,33,36)(28,38,39,30)(29,40,34,35);(10,16)(15,18)(17,23)(22,24)(26,33)(27,37)(28,39)(35,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28)(26,34)(27,38)(29,33)(30,37)(31,39)(32,35)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,21)(10,17,16,23)(11,19,13,20)(15,22,18,24)(25,29,31,34)(26,28,33,39)(27,35,37,40)(30,36,38,32);(1,4,5,6)(2,7,8,3)(9,13,14,11)(10,18,16,15)(12,19,21,20)(17,22,23,24)(25,30,31,38)(26,35,33,40)(27,28,37,39)(29,36,34,32)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623465);