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 |
| prime radical | #35210 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(11,20)(12,18)(13,15)(14,17)(16,19)(21,24)(22,23)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(9,11,16,14)(10,17,19,20)(12,23,13,21)(15,22,18,24);(9,12,16,13)(10,18,19,15)(11,21,14,23)(17,22,20,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,3,5,4)(2,7,6,8)(9,13,16,12)(10,15,19,18)(11,21,14,23)(17,22,20,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(1,4,5,3)(2,8,6,7)(9,14,16,11)(10,17,19,20)(12,23,13,21)(15,24,18,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,15)(10,12)(11,22)(13,19)(14,24)(16,18)(17,23)(20,21)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553119);