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 | #46093 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26)(27,28)(29,34)(30,35)(31,32)(33,40)(36,38)(37,39);(1,2)(3,7)(4,5)(6,8)(10,16)(11,12)(18,23)(19,20)(26,32)(27,33)(34,35)(36,37);(2,5)(6,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,31,33)(26,28,32,40)(29,36,30,37)(34,38,35,39);(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,33)(27,32)(29,38)(30,39)(31,40)(34,37)(35,36);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,30)(26,34,32,35)(27,36,33,37)(28,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,29)(26,35,32,34)(27,37,33,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8121941);