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 | |
4503 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)(11,18)(12,13)(14,16)(15,21)(17,23)(19,20)(22,24)(26,31)(27,33)(34,38)(35,40);(1,2,5,6)(3,7,8,4)(9,11,14,21)(10,15,16,18)(12,22,23,19)(13,20,17,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(2,6)(4,7)(10,16)(13,17)(15,18)(20,24)(29,37)(34,38)(35,40)(36,39);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,22)(13,20)(14,21)(16,18)(17,24)(19,23)(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)(10,17)(11,19)(13,16)(14,23)(15,20)(18,24)(21,22)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,4)(2,3)(5,7)(6,8)(9,13,14,17)(10,12,16,23)(11,20,21,24)(15,22,18,19)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8265695);