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 | #43102 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4690 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)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(2,6)(3,8)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(26,30)(31,37)(32,35)(38,39);(1,3)(2,4)(5,8)(6,7)(10,15)(13,22)(16,19)(20,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,27)(26,31)(28,33)(29,34)(30,37)(32,38)(35,39)(36,40)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=hi,e^2=hi,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, 8135733);