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 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4346 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)(3,7)(4,8)(5,6)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(25,26,31,35)(27,30,34,39)(28,36,38,32)(29,33,40,37);(2,6)(3,8)(9,11,15,20)(10,16,19,22)(12,14,21,23)(13,17,18,24)(25,27,31,34)(26,30,35,39)(28,37,38,33)(29,32,40,36);(1,3)(2,4)(5,8)(6,7)(11,20)(12,21)(16,22)(17,24)(25,28,31,38)(26,32,35,36)(27,33,34,37)(29,39,40,30);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,14)(13,22)(15,21)(16,18)(19,24)(20,23)(25,29,31,40)(26,33,35,37)(27,36,34,32)(28,30,38,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,14)(11,17)(12,22)(15,18)(16,21)(19,23)(20,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(9,14)(10,18)(11,21)(12,20)(13,19)(15,23)(16,24)(17,22)(25,30)(26,34)(27,35)(28,29)(31,39)(32,37)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619444);