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 | #50527 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4281 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)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,26)(27,28)(29,33)(30,32)(31,34)(35,36)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11,14,15)(10,12,17,19)(13,18,22,23)(16,20,21,24)(25,27)(26,29)(28,30)(31,35)(32,33)(34,37)(36,38)(39,40);(11,15)(12,19)(18,23)(20,24)(25,28,32,29)(26,33,30,27)(31,37,39,36)(34,35,38,40);(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,24)(20,23)(25,29,32,28)(26,33,30,27)(31,36,39,37)(34,35,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24)(25,30,32,26)(27,28,33,29)(31,38,39,34)(35,36,40,37);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,31,32,39)(26,34,30,38)(27,35,33,40)(28,36,29,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7980324);