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 | #50893 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4467 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,30,32)(27,31,35,34)(28,33,39,29)(36,37,38,40);(1,2,5,6)(3,7,4,8)(25,27)(26,31)(28,38)(29,37)(30,35)(32,34)(33,40)(36,39);(10,15)(13,22)(16,23)(20,24)(26,32)(27,35)(29,33)(36,38);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,30,39)(26,33,32,29)(27,36,35,38)(31,37,34,40);(1,3,5,4)(2,7,6,8)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,27)(26,34)(28,36)(29,37)(30,35)(31,32)(33,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29,30,33)(26,28,32,39)(27,37,35,40)(31,38,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,e^2=ghi,f^2=gi,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, 7998926);