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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,22)(12,13,20,21)(17,18,23,24)(25,26)(27,34)(28,33)(29,38)(30,31)(32,37)(35,36)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,19)(10,15,16,22)(12,17,20,23)(13,18,21,24)(26,31)(28,38)(32,34)(35,40);(1,3)(2,5)(4,8)(6,7)(10,16)(12,20)(15,22)(17,23)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(9,12)(10,13)(11,17)(14,20)(15,18)(16,21)(19,23)(22,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,30,38)(26,33,31,29)(27,35,37,40)(32,39,34,36);(1,2)(3,5)(4,7)(6,8)(9,13)(10,12)(11,18)(14,21)(15,17)(16,20)(19,24)(22,23)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9575795);