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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26,31,35)(27,36,39,32)(28,38,37,33)(29,34,40,30);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,20,21)(12,23,18,14)(13,24,17,19);(11,22)(14,23)(16,21)(19,24)(25,27,31,39)(26,32,35,36)(28,29,37,40)(30,33,34,38);(9,12)(10,17)(11,23)(13,20)(14,22)(15,18)(16,19)(21,24)(25,28)(26,33)(27,29)(30,36)(31,37)(32,34)(35,38)(39,40);(9,13,15,17)(10,18,20,12)(11,24,22,19)(14,21,23,16)(25,29)(26,34)(27,37)(28,39)(30,35)(31,40)(32,38)(33,36);(1,3,4,7)(2,6,5,8)(9,14,15,23)(10,19,20,24)(11,12,22,18)(13,21,17,16)(25,30,31,34)(26,29,35,40)(27,38,39,33)(28,36,37,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7558907);