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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,21)(18,22)(19,20)(23,24)(25,26,30,34)(27,33,29,31)(28,32,36,39)(35,40,37,38);(2,5)(3,7)(11,18)(16,22)(19,23)(20,24)(25,27,30,29)(26,31,34,33)(28,35,36,37)(32,38,39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,22)(19,23)(28,36)(32,39)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,14,17)(10,13,15,21)(11,19,18,23)(16,20,22,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,13)(10,17)(11,20)(12,15)(14,21)(16,23)(18,24)(19,22)(25,29,30,27)(26,33,34,31)(28,37,36,35)(32,40,39,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002699);