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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,21,23,17)(19,24,22,20)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,22,23,19)(17,20,21,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(3,8)(4,6)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,15)(17,22)(19,21)(23,24)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,23)(10,17,18,21)(11,19,16,22)(12,20,15,24)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8877823);