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 | |
4460 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,39)(37,40);(1,3,5,8)(2,6,4,7)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(10,11)(15,17)(16,18)(22,23)(26,28)(27,33)(32,36)(35,40);(1,4,5,2)(3,7,8,6)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,37,40,39);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,40)(34,37);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,34)(26,33,28,27)(29,37,38,39)(32,35,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=ghi,f^2=h,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, 7854357);