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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,21)(23,24)(25,26)(27,36)(28,29)(30,37)(31,35)(32,39)(33,40)(34,38);(1,2,5,3)(4,6,8,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,23,20,21)(25,27)(26,32)(28,37)(29,38)(30,33)(31,39)(34,40)(35,36);(2,3)(6,7)(10,16)(11,18)(17,20)(19,24)(25,28,31,40)(26,29,35,33)(27,37,39,34)(30,32,38,36);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,17)(14,16)(15,18)(19,21)(20,22)(23,24)(25,28)(26,33)(27,37)(29,35)(30,32)(31,40)(34,39)(36,38);(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,21,15,23)(25,29,31,33)(26,28,35,40)(27,38,39,30)(32,37,36,34);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,24)(22,23)(25,30,31,38)(26,34,35,37)(27,33,39,29)(28,32,40,36)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014709);