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 | #40768 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4234 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(25,26,30,32)(27,28,33,35)(29,31,38,40)(34,36,37,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,19)(17,24)(18,21)(27,33)(28,35)(34,37)(36,39);(1,2,6,5)(3,8,7,4)(10,15)(11,19)(17,24)(18,21)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40);(1,4)(2,7)(3,5)(6,8)(9,11,14,19)(10,16,15,22)(12,18,20,21)(13,17,23,24)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,27)(26,28)(29,37)(30,33)(31,39)(32,35)(34,38)(36,40);(1,5,6,2)(3,8,7,4)(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,29)(26,31)(27,34)(28,36)(30,38)(32,40)(33,37)(35,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9845074);