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 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,37,40,39);(1,2)(3,7)(4,8)(5,6)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(10,15)(11,18)(17,20)(19,24)(26,31)(27,34)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,37,36,39);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,33)(27,35)(29,39)(30,38)(31,36)(32,37)(34,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7824410);