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 | #45642 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(13,21)(18,22)(19,24)(20,23)(25,26,30,34)(27,33,29,31)(28,32,37,40)(35,39,36,38);(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,30,29)(26,31,34,33)(28,35,37,36)(32,38,40,39);(1,3,5,2)(4,7,8,6)(10,15)(12,16)(18,22)(20,23)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(1,4,5,8)(2,6,3,7)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,30,27)(26,33,34,31)(28,36,37,35)(32,39,40,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=gh,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, 8002943);