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 | |
4419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,15,19)(11,14,18,23)(12,22,21,16)(13,17,20,24)(25,26,31,34)(27,35,38,30)(28,29,33,32)(36,39,40,37);(1,3,6,4)(2,7,5,8)(9,11,15,18)(10,14,19,23)(12,13,21,20)(16,24,22,17)(25,27,31,38)(26,30,34,35)(28,36,33,40)(29,37,32,39);(9,12,15,21)(10,16,19,22)(11,20,18,13)(14,17,23,24)(25,28,31,33)(26,32,34,29)(27,36,38,40)(30,39,35,37);(2,5)(7,8)(9,11)(10,14)(12,20)(13,21)(15,18)(16,17)(19,23)(22,24)(25,26)(27,35)(28,32)(29,33)(30,38)(31,34)(36,37)(39,40);(1,4,6,3)(2,8,5,7)(9,13,15,20)(10,17,19,24)(11,21,18,12)(14,16,23,22)(25,29,31,32)(26,33,34,28)(27,37,38,39)(30,40,35,36);(1,5,6,2)(3,8,4,7)(9,14)(10,18)(11,19)(12,17)(13,16)(15,23)(20,22)(21,24)(25,30,31,35)(26,27,34,38)(28,39,33,37)(29,40,32,36)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,e^2=gi,f^2=g,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, 8352888);