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 | |
4581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,15,20,16)(12,13,22,24)(18,19,21,23)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(2,5)(6,7)(9,11,14,20)(10,16,17,15)(12,21,22,18)(13,19,24,23)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,22)(15,19)(16,23)(17,24)(20,21)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,19)(14,24)(15,18)(16,21)(17,22)(20,23)(25,29,30,38)(26,33,34,40)(27,35,32,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8665216);