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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,12,13,16)(14,17,22,24)(19,20,21,23)(25,26)(27,28)(29,37)(30,33)(31,36)(32,38)(34,40)(35,39);(1,2,5,6)(3,7,8,4)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,27,30,31)(26,28,33,36)(29,34,38,39)(32,35,37,40);(2,6)(4,7)(14,22)(17,24)(19,21)(20,23)(25,28,30,36)(26,31,33,27)(29,35,38,40)(32,39,37,34);(1,3)(2,4)(5,8)(6,7)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21);(9,13,15,11)(10,16,18,12)(14,21,22,19)(17,23,24,20)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,37)(36,40);(1,4)(2,3)(5,7)(6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,26)(27,28)(29,32)(30,33)(31,36)(34,35)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^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, 8088324);