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 | |
4592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,21)(13,23)(14,16)(15,17)(19,24)(20,22)(25,26)(27,33)(28,31)(29,34)(30,35)(32,38)(36,39)(37,40);(1,2,5,8)(3,4,7,6)(9,11,15,14)(10,16,17,18)(12,22,23,19)(13,20,21,24)(25,27)(26,29)(28,36)(30,34)(31,37)(32,39)(33,35)(38,40);(1,3)(2,6)(4,8)(5,7)(10,17)(12,23)(16,18)(19,22)(25,28,30,38)(26,31,35,32)(27,36,34,40)(29,37,33,39);(1,4,5,6)(2,7,8,3)(9,12)(10,13)(11,19)(14,22)(15,23)(16,24)(17,21)(18,20)(25,28)(26,32)(27,36)(29,39)(30,38)(31,35)(33,37)(34,40);(1,3)(2,4)(5,7)(6,8)(9,13)(10,12)(11,20)(14,24)(15,21)(16,22)(17,23)(18,19)(25,27)(26,33)(28,36)(29,35)(30,34)(31,39)(32,37)(38,40);(1,4)(2,7)(3,8)(5,6)(9,14,15,11)(10,16,17,18)(12,19,23,22)(13,20,21,24)(25,29,30,33)(26,34,35,27)(28,37,38,39)(31,40,32,36)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8623917);