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 | #44170 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,40)(36,38);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,18)(14,21)(16,17)(19,22)(23,24)(25,27)(26,31)(28,37)(29,36)(30,35)(32,33)(34,40)(38,39);(1,2,5,4)(3,6,8,7)(10,16)(13,23)(15,17)(18,24)(27,35)(29,37)(31,40)(33,39);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,19)(13,18)(14,21)(15,16)(20,22)(23,24)(25,28)(26,32)(27,29)(30,36)(31,39)(33,40)(34,38)(35,37);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,20)(13,15)(14,22)(16,24)(17,23)(19,21)(25,29)(26,33)(27,36)(28,35)(30,37)(31,32)(34,39)(38,40);(9,13,14,23)(10,19,16,20)(11,18,21,24)(12,15,22,17)(25,26)(27,31)(28,38)(29,33)(30,34)(32,36)(35,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 7987990);