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 | |
4358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(25,26)(27,33)(28,38)(29,34)(30,32)(31,35)(36,40)(37,39);(1,2,5,4)(3,6,7,8)(10,15)(11,20)(17,22)(18,21)(25,27,32,39)(26,33,30,37)(28,36,29,35)(31,38,40,34);(2,4)(6,8)(10,15)(13,24)(16,19)(17,22)(25,28,32,29)(26,34,30,38)(27,35,39,36)(31,33,40,37);(3,7)(6,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,29,32,28)(26,34,30,38)(27,36,39,35)(31,33,40,37);(1,3,5,7)(2,6,4,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,30,32,26)(27,37,39,33)(28,38,29,34)(31,36,40,35);(1,4,5,2)(3,8,7,6)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,31,32,40)(26,35,30,36)(27,38,39,34)(28,37,29,33)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8248997);