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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4664 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)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,22,24,21)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,39)(38,40);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,21)(20,24)(22,23)(25,27,31,29)(26,32,33,35)(28,38,39,36)(30,37,40,34);(1,3)(2,4)(5,8)(6,7)(12,20)(16,23)(19,24)(21,22)(26,33)(28,39)(32,35)(36,38);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,24)(18,23)(25,28)(26,34)(27,36)(29,38)(30,35)(31,39)(32,40)(33,37);(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,29,31,27)(26,35,33,32)(28,38,39,36)(30,37,40,34);(9,13)(10,17)(11,18)(12,21)(14,15)(16,19)(20,22)(23,24)(25,30)(26,36)(27,37)(28,32)(29,34)(31,40)(33,38)(35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641983);