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 | |
4682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,21)(13,23)(14,17)(15,18)(19,24)(20,22)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,39,40,36);(3,7)(6,8)(11,18)(15,16)(19,22)(20,24)(26,31)(28,32)(33,37)(36,39);(1,2,5,4)(3,6,7,8)(9,11,14,18)(10,15,17,16)(12,22,23,19)(13,20,21,24)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,19)(13,20)(14,18)(15,17)(21,24)(22,23)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,4,5,2)(3,8,7,6)(9,12)(10,13)(11,19)(14,23)(15,24)(16,20)(17,21)(18,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,13)(10,12)(11,20)(14,21)(15,22)(16,19)(17,23)(18,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038367);