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 | |
4428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,31,30)(27,34,29,32)(28,36,38,33)(35,39,37,40);(1,2,5,4)(3,6,7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(2,4)(6,8)(13,21)(16,23)(18,24)(20,22)(26,30)(32,34)(33,36)(39,40);(3,7)(6,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,3,5,7)(2,6,4,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,29,31,27)(26,34,30,32)(28,35,38,37)(33,39,36,40);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,30,31,26)(27,34,29,32)(28,36,38,33)(35,40,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8249083);