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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,22)(13,24)(14,18)(15,21)(16,23)(17,20)(25,26)(27,29)(28,33)(30,35)(31,36)(32,34)(37,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,17,22,24)(25,27,31,34)(26,32,36,29)(28,35,39,37)(30,33,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,16,18,22)(11,13,21,20)(15,24,19,17)(25,28)(26,33)(27,37)(29,38)(30,32)(31,39)(34,35)(36,40);(9,12)(10,16)(11,20)(13,21)(14,23)(15,17)(18,22)(19,24)(25,29,31,32)(26,34,36,27)(28,38,39,30)(33,35,40,37);(1,4,5,3)(2,8,6,7)(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,22,19,16)(25,30,31,38)(26,35,36,37)(27,33,34,40)(28,32,39,29);(9,10)(11,15)(12,16)(13,24)(14,18)(17,20)(19,21)(22,23)(25,26)(27,32)(28,33)(29,34)(30,37)(31,36)(35,38)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=hi,e^2=g,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, 7838762);