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 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(10,15)(11,17)(12,19)(20,24)(25,26,30,29)(27,32,34,31)(28,33,37,36)(35,39,40,38);(2,6)(3,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(27,34)(31,32)(35,40)(38,39);(1,3,5,7)(2,4,6,8)(25,27)(26,31)(28,35)(29,32)(30,34)(33,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,38)(37,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,37)(26,33,29,36)(27,35,34,40)(31,38,32,39);(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29,30,26)(27,32,34,31)(28,36,37,33)(35,39,40,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,e^2=h,f^2=h,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, 8344986);