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 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(10,15)(11,17)(12,19)(20,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40);(1,2,5,6)(3,4,8,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,27,30,37)(26,31,35,32)(28,34,38,40)(29,36,39,33);(1,3)(2,4)(5,8)(6,7)(25,27)(26,32)(28,34)(29,36)(30,37)(31,35)(33,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,39)(38,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,38)(26,33,35,36)(27,34,37,40)(29,32,39,31);(1,4,5,7)(2,8,6,3)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,38)(33,37)(35,40)Polycyclic presentation:
2^9|b^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8368891);