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 | #56378 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4203 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)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24)(27,32)(29,34)(35,36)(37,39);(2,5)(6,7)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27,30,32)(26,29,33,34)(28,36,38,35)(31,39,40,37);(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,28,30,38)(26,31,33,40)(27,35,32,36)(29,37,34,39);(1,2)(3,7)(4,5)(6,8)(9,13,14,18)(10,16,17,11)(12,21,22,24)(15,23,20,19)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,29)(26,32)(27,33)(28,37)(30,34)(31,36)(35,40)(38,39)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9686460);