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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,21)(13,18)(14,16)(15,22)(17,23)(20,24)(26,29)(31,34)(32,37)(39,40);(1,2,5,7)(3,4,8,6)(9,11,14,22)(10,15,16,19)(12,18,23,20)(13,21,24,17);(1,3,5,8)(2,6,7,4)(10,16)(13,24)(15,19)(17,21)(25,26,30,29)(27,31,35,34)(28,32,38,37)(33,39,36,40);(1,3,5,8)(2,6,7,4)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,27,30,35)(26,31,29,34)(28,36,38,33)(32,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,28,30,38)(26,32,29,37)(27,33,35,36)(31,39,34,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,29,30,26)(27,34,35,31)(28,37,38,32)(33,40,36,39)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275654);