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 | |
4515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(13,21)(18,22)(19,24)(20,23)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(2,3)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,3,5,2)(4,7,8,6)(10,15)(12,16)(18,22)(20,23)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,4)(2,6)(3,7)(5,8)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,38)(39,40);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002922);