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 | |
4557 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)(3,6)(4,7)(5,8)(12,20)(16,23)(19,24)(21,22)(26,32)(27,37)(28,36)(35,39);(1,3,5,7)(2,6,8,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,32)(27,33,37,40)(28,30,36,34)(29,35,38,39);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,31,37)(26,33,32,40)(28,38,36,29)(30,39,34,35);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,34)(27,38)(29,37)(30,32)(31,36)(33,35)(39,40);(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,29,31,38)(26,35,32,39)(27,28,37,36)(30,40,34,33);(1,4,5,6)(2,7,8,3)(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,30,31,34)(26,36,32,28)(27,39,37,35)(29,40,38,33)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=hi,f^2=gi,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, 8365979);