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 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,4)(13,21)(16,23)(19,22)(20,24)(25,26)(27,34)(28,33)(29,39)(30,32)(31,35)(36,37)(38,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,37,33,40);(1,3,5,4)(2,7,6,8)(9,11,14,15)(10,12,17,18)(13,19,21,22)(16,20,23,24)(25,28,31,39)(26,33,35,29)(27,36,30,38)(32,40,34,37);(1,2,5,6)(3,7,4,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,29)(26,28)(27,37)(30,40)(31,33)(32,36)(34,38)(35,39);(1,2)(3,7)(4,8)(5,6)(9,12)(10,15)(11,17)(13,20)(14,18)(16,22)(19,23)(21,24)(25,26)(27,32)(28,33)(29,39)(30,34)(31,35)(36,40)(37,38);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,40,33,37)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044775);