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 | #59099 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(12,16)(17,23)(20,24)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(26,32)(27,28)(34,39)(35,36);(10,15)(12,16)(17,23)(20,24)(25,27,31,28)(26,33,32,29)(30,36,38,35)(34,37,39,40);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,12)(13,21,22,18)(17,20,23,24)(25,28,31,27)(26,33,32,29)(30,35,38,36)(34,37,39,40);(1,4,5,3)(2,8,6,7)(9,12,14,16)(10,11,15,19)(13,20,22,24)(17,18,23,21)(25,29,31,33)(26,27,32,28)(30,37,38,40)(34,35,39,36);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7984619);