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 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24)(25,26)(27,35)(28,32)(29,39)(30,34)(31,36)(33,37)(38,40);(1,3,5,7)(2,4,8,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,39,40,33);(1,3)(2,6)(4,8)(5,7)(25,26,30,34)(27,31,36,35)(28,32,38,40)(29,33,37,39);(1,2,5,8)(3,4,7,6)(9,12,14,21)(10,16,18,24)(11,20,22,13)(15,17,19,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(1,3)(2,4)(5,7)(6,8)(9,13,14,20)(10,17,18,23)(11,21,22,12)(15,16,19,24)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,33,35,39);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,16)(13,23)(14,18)(17,20)(19,22)(21,24)(25,29)(26,33)(27,28)(30,37)(31,32)(34,39)(35,40)(36,38)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9633548);