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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,19)(12,22)(13,21)(14,15)(16,20)(17,23)(18,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,22)(18,21)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(2,5)(6,8)(10,15)(12,23)(16,19)(18,21)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,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, 8023114);