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 | #46423 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,16)(12,19)(13,21)(14,15)(17,22)(18,20)(23,24)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(2,5)(6,8)(10,15)(12,16)(17,21)(20,23)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8030168);