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 | #48363 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,19)(11,16,22,23)(12,14,21,17)(13,18,20,24)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(2,6)(3,7)(9,11,15,22)(10,16,19,23)(12,13,21,20)(14,18,17,24)(26,31)(28,32)(33,37)(36,39);(1,3)(2,4)(5,7)(6,8)(9,12,15,21)(10,17,19,14)(11,20,22,13)(16,18,23,24)(25,26,30,31)(27,32,35,28)(29,33,38,37)(34,39,40,36);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,13)(14,19)(15,21)(16,24)(18,23)(20,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,13)(10,18)(11,21)(12,22)(14,23)(15,20)(16,17)(19,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(9,14,15,17)(10,12,19,21)(11,18,22,24)(13,23,20,16)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961596);