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 | #51523 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,13,18,16)(12,21,22,14)(19,20,24,23)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2,5,6)(3,7,4,8)(9,11,15,18)(10,16,17,13)(12,19,22,24)(14,20,21,23);(1,3,5,4)(2,7,6,8)(10,17)(11,18)(12,22)(20,23)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(9,13)(10,11)(12,23)(14,19)(15,16)(17,18)(20,22)(21,24)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,3)(2,8,6,7)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=hi,f^2=gh,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, 7804895);