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 | #43112 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,38,40,39);(1,3,6,5)(2,4,8,7)(9,10,15,12)(11,16,13,18)(14,17,22,20)(19,23,21,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,4)(5,7)(6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4)(2,5)(3,8)(6,7)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24);(1,4)(2,3)(5,8)(6,7)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,5,6,3)(2,7,8,4)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613543);