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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,39)(37,38);(1,3)(2,4)(5,7)(6,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,28)(26,32,33,34)(29,36,40,38)(30,37,35,39);(2,6)(3,7)(13,21)(16,20)(18,24)(22,23)(26,33)(30,35)(32,34)(37,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,31,27)(26,34,33,32)(29,38,40,36)(30,39,35,37);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,35)(27,36)(28,38)(30,33)(31,40)(32,39)(34,37);(1,3)(2,4)(5,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,35)(26,29,33,40)(27,37,28,39)(32,36,34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8100888);