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 | #47927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,23)(14,18)(17,21)(19,24)(20,22)(25,26)(27,35)(28,30)(29,34)(31,33)(32,38)(36,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,20)(14,19)(16,17)(18,24)(22,23)(25,27,31,38)(26,32,33,35)(28,29,36,39)(30,37,40,34);(1,4,5,2)(3,7,8,6)(11,19)(13,21)(15,24)(17,23)(26,33)(30,40)(32,35)(34,37);(9,12)(10,16)(11,13)(14,20)(15,23)(17,24)(18,22)(19,21)(25,28)(26,30)(27,29)(31,36)(32,37)(33,40)(34,35)(38,39);(9,13)(10,17)(11,20)(12,19)(14,21)(15,16)(18,23)(22,24)(25,29)(26,34)(27,36)(28,38)(30,32)(31,39)(33,37)(35,40);(9,10)(11,15)(12,22)(13,17)(14,18)(16,20)(19,24)(21,23)(25,30)(26,28)(27,37)(29,32)(31,40)(33,36)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,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, 7984603);