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 | |
4735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,11)(15,17)(16,18)(22,23)(25,26)(27,28)(29,31)(30,32)(33,36)(34,35)(37,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27,31,33)(26,28,29,36)(30,34,39,40)(32,35,37,38);(2,7)(4,6)(12,19)(15,17)(20,24)(22,23)(26,29)(27,33)(32,37)(34,40);(1,3,5,8)(2,6,7,4)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,28)(26,27)(29,33)(30,38)(31,36)(32,40)(34,37)(35,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,26)(27,28,33,36)(30,37,39,32)(34,35,40,38);(1,3)(2,4)(5,8)(6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,32)(27,34)(28,35)(29,37)(31,39)(33,40)(36,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495557);