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 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,12)(14,18)(20,22)(26,31)(29,32)(33,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,10,15,16)(11,19,12,17)(13,18,24,14)(20,21,22,23);(2,5)(6,7)(9,11,15,12)(10,17,16,19)(13,20,24,22)(14,21,18,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(2,5)(6,7)(9,12,15,11)(10,17,16,19)(13,20,24,22)(14,23,18,21)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,20)(12,22)(14,16)(15,24)(17,23)(19,21)(25,28,30,38)(26,33,31,39)(27,35,34,36)(29,37,32,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,18)(10,13,16,24)(11,21,12,23)(17,20,19,22)(25,29,30,32)(26,34,31,27)(28,37,38,40)(33,36,39,35)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231188);