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 | |
4547 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,5)(6,7)(10,15)(11,19)(13,22)(20,23);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,18)(20,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,4,8)(2,6,5,7)(9,11,14,19)(10,16,15,18)(12,20,17,23)(13,21,22,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10243978);