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 | #35703 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4554 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,29,33)(27,30,35,31)(28,32,37,40)(34,38,36,39);(1,2,5,6)(3,7,8,4)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,18)(13,22)(19,23)(25,27,29,35)(26,31,33,30)(28,34,37,36)(32,39,40,38);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947482);