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 |
| prime radical | #49682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,14,15)(11,17,19,13)(12,16,21,22)(18,23,24,20)(26,28)(27,33)(32,36)(35,39);(1,3,5,8)(2,6,7,4)(10,15)(13,17)(16,22)(20,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(9,11)(10,13)(12,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4)(2,3)(5,6)(7,8)(9,12,14,21)(10,16,15,22)(11,18,19,24)(13,20,17,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,11)(12,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=hi,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, 8099628);