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 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(11,17)(15,22)(18,20)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3,6,7)(2,4,8,5)(10,14)(11,17)(16,19)(18,20)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(1,2)(3,4)(5,7)(6,8)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,29,30,38)(26,33,34,40)(27,35,32,37)(28,36,31,39)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^2=gi,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, 8838190);