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 |
| prime radical | #38865 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,18,16,12)(14,17,23,21)(19,22,24,20)(26,31)(28,35)(32,40)(36,38);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,22)(23,24)(26,31)(27,33)(32,40)(34,37);(1,3,5,6)(2,4,7,8)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,18)(10,16,13,11)(14,22,23,20)(17,19,21,24)(25,27)(26,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,10)(11,18,16,12)(14,17,23,21)(19,20,24,22)(25,28)(26,27)(29,38)(30,35)(31,33)(32,37)(34,40)(36,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,23)(10,17,13,21)(11,19,16,24)(12,20,18,22)(25,29,30,39)(26,32,31,40)(27,34,33,37)(28,36,35,38)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8572510);