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 | #38908 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,15,18)(11,14,17,23)(12,13,16,22)(19,20,21,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(2,7)(3,4)(9,11)(10,14)(12,19)(13,20)(15,17)(16,21)(18,23)(22,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,3,6,4)(2,8,7,5)(11,17)(14,23)(19,21)(20,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,4,6,3)(2,5,7,8)(9,12)(10,13)(11,19)(14,20)(15,16)(17,21)(18,22)(23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,5)(2,3)(4,7)(6,8)(9,13)(10,16)(11,20)(12,18)(14,21)(15,22)(17,24)(19,23)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38);(9,14,15,23)(10,17,18,11)(12,20,16,24)(13,21,22,19)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231305);