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 | #40725 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,34)(28,33)(29,38)(30,32)(31,35)(36,40)(37,39);(1,2,6,7)(3,5,4,8)(11,18)(14,15)(19,24)(21,22)(25,27,31,30)(26,32,35,34)(28,36,29,37)(33,39,38,40);(1,3,6,4)(2,5,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,28,31,29)(26,33,35,38)(27,36,30,37)(32,39,34,40);(1,4,6,3)(2,5,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,29,31,28)(26,33,35,38)(27,37,30,36)(32,39,34,40);(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,30,31,27)(26,34,35,32)(28,37,29,36)(33,40,38,39);(1,5)(2,4)(3,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,26)(27,32)(28,33)(29,38)(30,34)(31,35)(36,39)(37,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8337827);