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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(10,15)(12,18)(16,22)(20,24)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,38,40,39);(2,3)(6,7)(11,19)(16,22)(20,24)(21,23)(27,34)(31,33)(35,40)(38,39);(1,2,5,3)(4,6,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,23)(21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(1,4,5,8)(2,6,3,7)(9,12,14,18)(10,17,15,13)(11,20,19,24)(16,23,22,21)(25,28,30,37)(26,32,29,36)(27,35,34,40)(31,38,33,39);(9,13,14,17)(10,18,15,12)(11,21,19,23)(16,24,22,20)(25,29,30,26)(27,31,34,33)(28,36,37,32)(35,38,40,39)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087551);