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 | #2403 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,18)(11,19,22,16)(12,23,20,14)(13,17,24,21)(25,26)(27,32)(28,29)(30,34)(31,35)(33,40)(36,39)(37,38);(1,3,5,8)(2,6,7,4)(9,11,15,22)(10,16,18,19)(12,24,20,13)(14,21,23,17)(27,35)(31,32)(36,38)(37,39);(1,3)(2,4)(5,8)(6,7)(12,20)(13,24)(14,23)(17,21)(25,27,30,35)(26,31,34,32)(28,38,33,36)(29,39,40,37);(1,4)(2,3)(5,6)(7,8)(9,12)(10,14)(11,13)(15,20)(16,17)(18,23)(19,21)(22,24)(25,27)(26,32)(28,38)(29,37)(30,35)(31,34)(33,36)(39,40);(9,13)(10,17)(11,20)(12,22)(14,19)(15,24)(16,23)(18,21)(25,28,30,33)(26,29,34,40)(27,36,35,38)(31,37,32,39);(9,14)(10,12)(11,21)(13,19)(15,23)(16,24)(17,22)(18,20)(25,29)(26,33)(27,37)(28,34)(30,40)(31,38)(32,36)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966050);