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 | |
4307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27)(26,32)(28,38)(29,39)(30,34)(31,40)(33,35)(36,37);(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,31,34)(26,33,36,29)(27,38,40,30)(32,35,37,39);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,29,31,33)(26,34,36,28)(27,39,40,35)(30,37,38,32);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,30,31,38)(26,35,36,39)(27,34,40,28)(29,32,33,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759974);