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 | #35232 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(2,6)(3,7)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(1,3)(2,4)(5,7)(6,8)(10,17)(11,20)(14,21)(18,23)(25,27,31,28)(26,33,30,32)(29,35,38,36)(34,40,37,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8178432);