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 | #36677 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4244 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)(9,10,14,18)(11,16,21,15)(12,13,17,23)(19,22,24,20)(25,26,31,27)(28,35,38,32)(29,33,30,34)(36,40,37,39);(2,5)(3,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(26,27)(32,35)(33,34)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,18,23)(11,19,21,24)(15,20,16,22)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(1,3)(2,6)(4,7)(5,8)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,11)(10,16)(12,19)(13,22)(14,21)(15,18)(17,24)(20,23)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(9,13)(10,17)(11,20)(12,18)(14,23)(15,24)(16,19)(21,22)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=hi,f^2=hi,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, 8004985);