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 | |
4803 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26)(27,31)(28,33)(29,35)(30,32)(34,36)(37,39)(38,40);(3,7)(6,8)(10,15)(11,20)(17,21)(18,24)(25,27)(26,31)(28,35)(29,33)(30,36)(32,34)(37,38)(39,40);(1,2,5,4)(3,6,7,8)(10,15)(12,22)(16,19)(18,24)(26,32)(27,36)(33,40)(35,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,23)(22,24)(25,28)(26,33)(27,37)(29,34)(30,38)(31,39)(32,40)(35,36);(1,4,5,2)(3,8,7,6)(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,19,23,16)(25,27)(26,34)(28,37)(29,33)(30,36)(31,32)(35,38)(39,40);(9,13)(10,18)(11,21)(12,19)(14,23)(15,24)(16,22)(17,20)(25,29)(26,35)(27,33)(28,31)(30,39)(32,37)(34,38)(36,40)Polycyclic presentation:
2^9|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, 8442137);