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 | #45770 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,23)(21,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,37)(36,40)(38,39);(3,7)(6,8)(9,11)(10,16)(12,21)(13,18)(14,17)(15,20)(19,23)(22,24)(26,27)(29,37)(32,34)(36,38);(1,2)(3,6)(4,5)(7,8)(11,20)(12,18)(14,22)(16,23)(25,27,31,26)(28,34,30,32)(29,35,37,33)(36,40,38,39);(9,12,15,18)(10,14,19,22)(11,21,20,13)(16,17,23,24)(25,28,31,30)(26,32,27,34)(29,36,37,38)(33,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,21)(10,17,19,24)(11,12,20,18)(14,16,22,23)(25,29,31,37)(26,33,27,35)(28,36,30,38)(32,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,22)(20,24)(21,23)(25,30,31,28)(26,34,27,32)(29,38,37,36)(33,40,35,39)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839592);