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 | |
4278 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)(3,7)(4,6)(5,8)(9,10,15,13)(11,18,12,16)(14,21,22,17)(19,23,20,24)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,3,4,8)(2,5,6,7)(11,12)(14,22)(16,18)(17,21)(27,35)(29,39)(31,40)(33,36);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,16,12,18)(14,21,22,17)(19,23,20,24)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,14,15,22)(10,17,13,21)(11,19,12,20)(16,23,18,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235387);