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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24);(3,7)(6,8)(10,15)(11,18)(17,24)(19,21)(25,26)(27,33)(28,29)(30,34)(31,37)(32,38)(35,39)(36,40);(1,2,5,4)(3,6,7,8)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,28,30,38)(26,32,34,29)(27,35,37,40)(31,39,33,36);(1,4,5,2)(3,8,7,6)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,27)(26,33)(28,35)(29,39)(30,37)(31,34)(32,36)(38,40);(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,29,30,32)(26,28,34,38)(27,36,37,39)(31,35,33,40)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8608486);