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 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,20)(14,16)(15,19)(17,23)(18,24)(21,22)(26,31)(27,33)(34,38)(35,40);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,38)(39,40);(2,6)(4,7)(10,16)(13,15)(17,20)(22,24)(29,37)(34,38)(35,40)(36,39);(2,6)(3,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,15,24)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,4)(2,3)(5,7)(6,8)(9,13,14,15)(10,11,16,19)(12,22,23,24)(17,18,20,21)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|b^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8295979);