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 | |
4398 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)(11,15)(12,17)(13,19)(14,16)(18,20)(21,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(3,7)(6,8)(9,11)(10,15)(12,19)(13,17)(14,20)(16,18)(21,22)(23,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,2,5,4)(3,6,7,8)(10,16)(11,20)(17,24)(19,21)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,21)(13,18)(14,22)(15,23)(16,24)(19,20)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4,5,2)(3,8,7,6)(9,11)(10,18)(12,21)(13,17)(14,20)(15,16)(19,22)(23,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38);(9,13)(10,19)(11,17)(12,15)(14,23)(16,21)(18,22)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442140);