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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40);(2,3)(6,7)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,36)(31,32)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(25,27)(26,31)(28,37)(29,38)(30,36)(32,35)(33,39)(34,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,27)(26,32)(28,37)(29,39)(30,36)(31,35)(33,38)(34,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,28,30,34)(26,33,35,29)(27,37,36,40)(31,39,32,38);(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,29,30,33)(26,34,35,28)(27,38,36,39)(31,40,32,37)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,e^2=h,f^2=h,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, 8344976);