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 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,15,16)(11,17,12,21)(13,22,23,18)(14,19,24,20)(26,30)(27,33)(28,36)(38,40);(2,3)(6,7)(10,16)(13,23)(17,21)(19,20)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,3,5,2)(4,7,8,6)(9,11,15,12)(10,17,16,21)(13,20,23,19)(14,18,24,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,5,8)(2,6,3,7)(9,12,15,11)(10,17,16,21)(13,19,23,20)(14,18,24,22)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(9,13)(10,18)(11,20)(12,19)(14,21)(15,23)(16,22)(17,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(9,14,15,24)(10,19,16,20)(11,18,12,22)(13,21,23,17)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8403776);