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 | |
4378 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)(4,8)(6,7)(11,17)(15,22)(18,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(2,7)(3,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(26,31)(28,32)(34,39)(36,40);(1,3,6,8)(2,4,7,5)(10,14)(11,17)(16,19)(18,20)(25,27,30,33)(26,32,31,28)(29,37,38,35)(34,36,39,40);(1,4,6,5)(2,3,7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,39)(37,38);(1,2)(3,4)(5,8)(6,7)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,5,6,4)(2,8,7,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,29,30,38)(26,34,31,39)(27,35,33,37)(28,36,32,40)Polycyclic presentation:
2^9|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, 8838188);