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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(26,31)(27,34)(28,36)(37,38);(1,3,5,8)(2,6,4,7)(25,26,30,31)(27,32,34,29)(28,33,36,39)(35,37,40,38);(10,15)(11,16)(17,22)(18,23)(25,27,30,34)(26,29,31,32)(28,35,36,40)(33,38,39,37);(1,2,5,4)(3,6,8,7)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,37)(36,40)(38,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,39)(32,37)(34,40);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,32)(26,27,31,34)(28,37,36,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,f^2=ghi,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, 7805593);