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 | #37716 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4476 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,30,34)(27,31,36,40)(28,37,38,32)(29,33,35,39);(1,3,5,8)(2,6,4,7)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,33,37,39);(10,15)(11,17)(16,22)(18,23)(25,28,30,38)(26,32,34,37)(27,29,36,35)(31,39,40,33);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,33)(34,37)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,26)(27,31)(28,32)(29,39)(30,34)(33,35)(36,40)(37,38);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,37,40,32)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7853854);