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 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,16,18,12)(14,21,23,17)(19,20,22,24)(25,26,32,36)(27,37,39,33)(28,31,34,29)(30,35,40,38);(1,2,5,7)(3,4,8,6)(9,11)(10,16)(12,13)(14,22)(15,18)(17,20)(19,23)(21,24)(25,27,32,39)(26,33,36,37)(28,30,34,40)(29,35,31,38);(1,3,5,8)(2,6,7,4)(10,13)(11,18)(17,21)(19,22)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35)(37,38)(39,40);(1,3,5,8)(2,6,7,4)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,22)(23,24)(25,29,32,31)(26,34,36,28)(27,38,39,35)(30,37,40,33);(1,3)(2,4)(5,8)(6,7)(9,13,15,10)(11,12,18,16)(14,21,23,17)(19,20,22,24)(25,30)(26,35)(27,34)(28,39)(29,37)(31,33)(32,40)(36,38);(1,4)(2,3)(5,6)(7,8)(9,14,15,23)(10,17,13,21)(11,19,18,22)(12,20,16,24)(25,31,32,29)(26,28,36,34)(27,35,39,38)(30,33,40,37)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=hi,e^2=gh,f^2=h,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, 10275783);