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 | #36676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(12,20)(15,22)(18,24)(21,23)(25,26,31,34)(27,35,36,28)(29,32,30,33)(37,40,38,39);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(27,36)(28,35)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,28)(29,37)(30,38)(31,36)(32,39)(33,40)(34,35);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,28)(26,27)(29,39)(30,40)(31,35)(32,37)(33,38)(34,36);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,31,30)(26,32,34,33)(27,37,36,38)(28,39,35,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,30,31,29)(26,33,34,32)(27,38,36,37)(28,40,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8005010);