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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4287 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)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26)(27,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(17,21)(20,23)(26,27)(32,35)(33,34)(39,40);(10,15)(11,18)(17,21)(19,24)(25,27,31,26)(28,35,38,32)(29,34,30,33)(36,40,37,39);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28,31,38)(26,32,27,35)(29,36,30,37)(33,39,34,40);(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,31,30)(26,33,27,34)(28,36,38,37)(32,39,35,40);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30,31,29)(26,34,27,33)(28,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549308);